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September 5

Project Euler question

There is a Project Euler problem (whose number I am not quoting so the answer is not given away directly here), where one needs integer solutions to 2b^2 - 2b - n^2 + n = 0. b=3 and n=4 is one solution to this equation. The equations b' = 3b + 2n - 2 and n' = 4b +3n - 3 generate b' and n' which are also solutions to the equation. Two questions please: (1) How do I prove that the b' and n' equations are valid? (2) How does one do Diophantine magic and generate those equations in the first place? -- SGBailey (talk) 08:46, 5 September 2012 (UTC)[reply]

The equation is equivalent to 2b(b − 1) = n(n − 1). Now it becomes obvious b = n = 1 is another solution... --CiaPan (talk) 09:22, 5 September 2012 (UTC)[reply]

I know the equations work, I've got a dozen solutions and could have a dozen more if I wanted them (they get quite big...). But I'm interested in the proof and creation of the "next set of solutions" equations. Yes 1,1 is a solution. So is 0,1. But so is 3,4. -- SGBailey (talk) 09:36, 5 September 2012 (UTC)[reply]

I don't know where they come from, but it's simple enough to show that they work: just plug your formulas for b' and n' into the original equation and simplify. You'll get 2b^2 - 2b - n^2 + n = 0. Under the assumption that you started with a solution, this shows that b' and n' are a solution.--121.73.35.181 (talk) 10:25, 5 September 2012 (UTC)[reply]

(1) solved. You are right thanks - when I tried that previously, I went wrong; but it worked ok second time. So (2) where did the generating equations come from??? -- SGBailey (talk) 11:26, 5 September 2012 (UTC)[reply]

Convergents (p,q) to the continued fraction expansion of sqrt(2) are (1,1), (3,2), (7,5), (17,12), (41,29) etc (see Pell number). Alternate convergents (1,1), (7,5), (41,29) etc. satisfy

p^{2}=2q^{2}-1

Successive convergents in this sequence are related by the recurrence relations

p'=4q+3p\quad q'=3q+2p

Use the transform

n={\frac {p+1}{2}}\quad b={\frac {q+1}{2}}

and you get the (n,b) sequence (1,1), (4,3), (21, 15) etc. which satisfies

(2n-1)^{2}=2(2b-1)^{2}-1\Rightarrow 4n^{2}-4n+1=8b^{2}-8b+1\Rightarrow n^{2}-n=2b^{2}-2b

with the recurrence relations

n'=4b+3n-3\quad b'=3b+2n-2

Gandalf61 (talk) 12:08, 5 September 2012 (UTC)[reply]

Thank you (I think). I didn't understand most of that, but I'll study it for a few days and see if it makes any sense then. -- SGBailey (talk) 13:11, 5 September 2012 (UTC)[reply]

Complex variables inequality

Hi. I'm working in a book, and looking at a claim that goes as follows: For $|e^{z}-1|\leq {\frac {1}{2}}$ , we have ${\frac {1}{2}}|z|\leq |e^{z}-1|\leq 2|z|$ . I've checked this by doing a change of variables $\zeta =e^{z}-1$ , parameterizing the circle $|\zeta |={\frac {1}{2}}$ , and checking that ${\frac {1}{2}}|\log(\zeta +1)|\leq |\zeta |={\frac {1}{2}}\leq 2|\log(\zeta +1)|$ all around the circle (for the principal branch). I checked just by graphing the left and right as functions of a real variable, and sure enough, they stayed away from ${\frac {1}{2}}$ . This is terribly awkward, though. Does anyone know an easier way to see this? Thanks in advance. -GTBacchus^(talk) 19:54, 5 September 2012 (UTC)[reply]

Unless I’m missing something, the first part of the inequality is false.

I will type * for times, and ^ for ”to the power of”.

Because, suppose that z = 2pi*i. Then e^z = e^(2pi*i) = cos(2pi) + i*sin(2pi) = 1 + 0 = 1. So then |e^z - 1| = 0, which is less than 1/2. So z = 2pi*i satisfies the assumption.

But then the first part of the inequality, (1/2)|z| <= |e^z -1|, becomes (1/2)(2pi) <= 0, or pi <= 0, which is just not true. Cardamon (talk) 07:00, 8 September 2012 (UTC)[reply]

Yeah, I noticed that too. The inequality applies for z close to 0, not close to other multiples of 2pi*i. It can be fixed by adding the condition that |z|<1 or something.

I seem to have it figured out. The trick is to do the substitution

\zeta =e^{z}-1

, and assume that

|\zeta |\leq {\frac {1}{2}}

; this obviates the problem you mention. Then you really want to show that

{\frac {1}{2}}|\zeta |\leq |\log(\zeta +1)|\leq 2|\zeta |

. This can be accomplished by playing with the series expansion of

\log(\zeta +1)

and the triangle inequality. -GTBacchus^(talk) 01:59, 11 September 2012 (UTC)[reply]

September 6

Implementing an algorithm to generate Proth Primes

Hello all. I'm trying to implement an efficient Proth Prime generator using Proth's theorem, but there are a few concepts that I am a little unclear about. Primarily:

- How many values of 'a' (on a logarithmic scale, perhaps) must be used to make the primality test deterministic?

- How should a given 'a' be chosen? Start at, say, 2 and then simply increment, or what?

- Would it be more efficient to first calculate the Jacobi symbol and test that 'a' is a quadratic nonresidue, or simply select 'a' by some other means?

Apologies for the ignorance, but I'm a computer programmer, not a mathematician! Thanks.

66.87.80.180 (talk) 00:19, 6 September 2012 (UTC)[reply]

For the first question, the complexity of finding a such that (a|p) = −1 deterministically is a difficult open problem. Assuming the generalized Riemann hypothesis for quadratic Dirichlet characters, one can always find an a ≤ 2(ln p)² which either has this property or is a proper divisor of p. On the other hand, unconditionally known bounds are essentially exponential, IIRC they are of the form p^ε for some constant ε.

For the second question, the best way is to chose a randomly. If you want to do it deterministically, I’m not aware of any advantage of doing it any other way than testing small a successively as you write, except that in this case, it suffices to test only prime a.

I don’t quite understand the third question, what other means do you have in mind?—Emil J. 11:45, 6 September 2012 (UTC)[reply]

Okay, thanks. Taking all of that into consideration, it seems that I might as well just stick with the Miller-Rabin test and abandon Proth Primes altogether then, as it has a much lower known lower bound (and an even lower heuristic lower bound) besides being suitable for primes of any form anyway. Cheers!

66.87.126.188 (talk) 14:50, 6 September 2012 (UTC)[reply]

Polynomial equations

hi, i just wonder what could b so wrong in solving math polynomial eqs. by choosing 2 arbitrary values 4 x, x1 n x2 then compute xm=(x1+x2)/2, wrinting P(x)=P(x-xm)+P(xm) then P(x1)=..similary, P(x2)=similary then somthing like P(y-a)=P(y+a)=0 y=xm a=xm-x1

P 4 ur delight, anyway LOL

Florin, Romania — Preceding unsigned comment added by 93.118.212.93 (talk) 13:02, 6 September 2012 (UTC)[reply]

Because for a general polynomial P, P(x) does not equal P(x-xm)+P(xm); for instance,

(x-x_{m})^{2}+x_{m}^{2}=x^{2}-2xx_{m}+2x_{m}^{2}

, which is not x^2 unless

x=x_{m}

which is unlikely since you have effectively taken xm arbitrary. Straightontillmorning (talk) 14:42, 6 September 2012 (UTC)[reply]

It seems like the OP might be groping for something like Newton's method. Looie496 (talk) 17:05, 6 September 2012 (UTC)[reply]

More on "Project Euler question" from Sept 5

Ok, with "Convergents (p,q) to the continued fraction expansion of sqrt(2)", it is sqrt(2) because the the equation has the form p^2 = 2*q^2 - 1. Had it been p^2 = 99*q^2 -1 it would have been sqrt(99) I presume.

It was said "Alternate convergents (1,1), (7,5), (41,29) etc. satisfy p^2 = 2*q^2 - 1". This is true by inspection, but how did you get alternates in the first place? Had I thought to do such a thing, I would have tried them all and given up when 9 = 3^2 != 2*2^2 - 1 = 7.

I now follow the mechanism that have been used here, but have no feel as to why/how it was decided theat these were good tools / transforms to use. What points in that direction? -- SGBailey (talk) 15:06, 6 September 2012 (UTC)[reply]

What points in that direction ? Mostly previous experience, having seen similar problems before and played around with continued fractions. Starting with

2b^{2}-2b=n^{2}-n

a natural first step is to complete the square on both sides, giving

2\left(b-{\frac {1}{2}}\right)^{2}-{\frac {1}{2}}=\left(n-{\frac {1}{2}}\right)^{2}-{\frac {1}{4}}

Multiplying both side by 4 to get rid of the fractions gives

2(2b-1)^{2}-1=(2n-1)^{2}

So if we set

p=2n-1\quad q=2b-1

then we want to find integers p and q such that

p^{2}=2q^{2}-1

\Rightarrow {\frac {p}{q}}\approx {\sqrt {2}}

which suggests we are looking for rational approximations to sqrt(2). The convergents to the continued fraction expansion of sqrt(2) are one such series of approximations, and I happened to know that they satisfy

p^{2}=2q^{2}\pm 1

with alternating signs. So it was then just a case of working out the recurrence relations between alternate convergents, and then translating this back into recurrence relations in terms of b and n instead of p and q. Gandalf61 (talk) 15:56, 6 September 2012 (UTC)[reply]

Many thanks. That made sense :-) -- SGBailey (talk) 18:54, 6 September 2012 (UTC)[reply]

surjective Homomorphism

Hi. Is it true that any two surjective homomorphisms $S_{4}\rightarrow S_{3}$ have the same kernel? By the correspondence theorem there exists a bijection between subgroups of $S_{4}$ containing the kernel of SOME homomorphism and the subgroups of $S_{3}$ but does this imply that this kernel is the same for any homomorphism?--150.203.114.14 (talk) 23:07, 6 September 2012 (UTC)[reply]

According to the symmetric group page, S₄ has only one normal subgroup of order 4 (which I think is the one consisting of the 2,2 cycle types). In that case, that's the only possible kernel of size 4 of a group homomorphism from S₄. Rckrone (talk) 02:28, 7 September 2012 (UTC)[reply]

This is easily seen from the class equation, where I've put the lengths of the cycles in each conjugacy class in parentheses: 24 = 1 + 3 (2,2) + 6 (2) + 8 (3) + 6 (4). So there's only one way to collect conjugacy classes into a subset with 4 elements, and that just happens to be a subgroup. --COVIZAPIBETEFOKY (talk) 11:36, 7 September 2012 (UTC)[reply]

September 7

Hyperbolic Trig?

While it may not be quite the same, I noticed that there are articles for both Trigonometry in a Planar concept as well as spherical geometry. Is there an inverted concept to spherical geometry in Hyberbolic spaces (and this is *not* as far as I can tell related to the Hyperbolic trig functions)Naraht (talk) 12:47, 7 September 2012 (UTC)[reply]

If you take a valid formula in spherical geometry and multiply all the lengths by i, you get a valid formula in hyperbolic geometry. A trig function of an imaginary argument becomes a hyperbolic function of a real argument (multiplied by some power of i).

The easiest way for me to understand hyperbolic geometry is with vectors in Minkowski space: (it,x,y,z). Again, the formulae are analogous to those of geometry on the unit hypersphere

w^{2}+x^{2}+y^{2}+z^{2}=1

, but the hyperbolic plane is represented by a unit hyperboloid

-t^{2}+x^{2}+y^{2}+z^{2}=-1

. —Tamfang (talk) 03:37, 8 September 2012 (UTC)[reply]

Rocket fin alignment

I have a rocket that has three sets of three fins, but they are not aligned properly. So I traced the circumference of the rocket onto a piece of paper. And I think that if I trisect the circle into 3 60 degree sections, I can use that as my guide to align the fins properly. But the problem is that I cannot find the center of this circle. So how do I trisect that circle into 3 60 degree sections with a pencil, compass, protractor, and ruler at my disposal? Legolover26 (talk) 14:28, 7 September 2012 (UTC)[reply]

It is easy to construct a diameter: pick two points on the circle. Open the compass to some big enough distance. Draw two arcs centred on the chosen points. The line through the points where the arcs cross is a diameter. Do this two or three times with different points to get more diameters which of course meet at the centre.--JohnBlackburne^words_deeds 15:23, 7 September 2012 (UTC)[reply]

Thank you very much! Legolover26 (talk) 16:03, 7 September 2012 (UTC)[reply]

Also note that 3 evenly spaced fins on a 360° circle will be 120° apart, not 60°. StuRat (talk) 16:09, 7 September 2012 (UTC)[reply]

Yea, I noticed that. I don't know why i said that. Maybe because the three angles of a triangle add up to 180 degrees. — Preceding unsigned comment added by Legolover26 (talk • contribs) 16:52, 7 September 2012 (UTC)[reply]

Here is another method you can use if you can't easily trace the rocket (let's say it has an engine sticking out the bottom):

1) Mark a spot on the edge of the rocket. This will be your first fin position.

2) Make a measuring tape by cutting a long strip of paper (if you already have a measuring tape, all the better, with the softer tailor's type being preferred over the stiff metal ones).

3) Wrap the tape around the rocket, starting at the mark, and extending until you return to the mark. Record this position on the tape, as well (or just note the measurement, if the measuring tape is numbered).

4) Then divide that by three, using the ruler to measure, if necessary (a ruler divided into tenths of an inch or mm is better than one divided into eights of an inch, as that makes for some ugly math).

5) Return the measuring tape to the rocket and mark those 2 additional fin locations.

One hint for using any ruler or measuring tape is to start at 1 inch (or cm), not at 0, since the end is often less accurate. Just remember to subtract 1 from your measurement). Some professional rulers don't start right at the end, to avoid this problem. StuRat (talk) 19:36, 7 September 2012 (UTC)[reply]

Axiom of empty set

The Zermelo-Fraenkel axioms do not include the axiom of empty set; instead the article contains the line:

The axiom of specification can be used to prove the existence of the empty set ... once the existence of at least one set is established (see above).

The axiom of infinity in the same article uses the fact that the empty set has already been defined. Is it because we implicitly assume that the domain of discourse contains at least 1 set and so by the axiom of specification the empty set?-Shahab (talk) 15:09, 7 September 2012 (UTC)[reply]

The axiom of infinity can be rephrased so as not to actually assume that any of the sets named actually exist. For example, to assert that the empty set is in X, simply say that for every y, if y has no elements, then y is an element of X. Then the axiom of infinity a priori only asserts the existence of a set meeting this description, without actually asserting that it contains anything. Then the existence of a set implies the existence of the empty set by the axiom of specification. --COVIZAPIBETEFOKY (talk) 16:25, 7 September 2012 (UTC)[reply]

Name of set

The Abel-Ruffini theorem article says "The theorem says that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational constants, and repeatedly forming sums, differences, products, quotients, and radicals (n-th roots, for some integer n) of previously obtained numbers."

What do you call the set of numbers that are statable by a finite number of sums, differences, products, quotients and radicals of integers? Is there a common name for this? RJFJR (talk) 16:37, 7 September 2012 (UTC)[reply]

"Numbers defined by radicals"? — Quondum 18:20, 7 September 2012 (UTC)[reply]

Sounds like irrational numbers, to me, unless we include roots of negative numbers, in which case we have complex numbers. StuRat (talk) 18:29, 7 September 2012 (UTC)[reply]

They will be irrational, but not all irrational numbers fit that description. The square root of two, for instance, is irrational but can be described in terms of radicals (I just described it that way). --Tango (talk) 22:30, 8 September 2012 (UTC)[reply]

Shouldn't your example be an irrational number which can't be described in terms of "repeatedly forming sums, differences, products, quotients, and radicals" to prove your point ? I would think any irrational number could be defined in that way, if we allow for an infinite series. However, when adding the restriction that it be a finite series, then you may well be right. StuRat (talk) 00:40, 9 September 2012 (UTC)[reply]

There are more irrationals (aleph1) than there are formulas constructable in a finite number of symbols. I was wondering if they might be related to the computable numbers. RJFJR (talk) 18:44, 7 September 2012 (UTC)[reply]

Just have to jump in here — there are

2^{\aleph _{0}}

irrationals. This may or may not equal

\aleph _{1}

— the statement that these are equal is the continuum hypothesis, and its truth value is not known.

(But it is certainly true that there are at least

\aleph _{1}

irrationals, and that's a possible reading of your claim, so apologies if that's what you meant.) --Trovatore (talk) 20:32, 7 September 2012 (UTC)[reply]

Nope. Pi is a computable number, but not a number defined by radicals (the latter set is presumably a proper subset of the former). — Quondum 19:05, 7 September 2012 (UTC)[reply]

You're right. And I now see where you liked Numbers defined by radicals to and that is the answer. Thank you. RJFJR (talk) 20:24, 7 September 2012 (UTC)[reply]

"The maximal solvable extension of Q", or "Q^solv" are names. You might have been thinking of constructible numbers which form a subfield, not computable numbers.John Z (talk) 20:55, 7 September 2012 (UTC)[reply]

I suspect that "Numbers defined by radicals" is a strictly informal term (perhaps "Numbers contructible from radicals" would be better). John's suggestion would be unambiguous (though I'd have to rely on him that it is in fact the same set; for example, there are solutions to the cubic and quartic equations that I'd assume are in Q^solv but are not constructible from only real radicals ~~of real rumbers~~ and integers). — Quondum 06:33, 8 September 2012 (UTC)[reply]

I thought it was called the surd field? Huh, that's red. I know I've seen it somewhere. --Trovatore (talk) 08:40, 8 September 2012 (UTC)[reply]

Surd field sounds like a good, snappy, short name, but a quick google seems to show that it used for what we are calling constructible numbers, and should probably redirect there. Quondum, I think you are referring to the fact that the solutions to say a rational cubic with 3 real roots may involve radicals of complex numbers, but by the cubic & quartic formulas they sit in Q^solv. Abel-Ruffini just says that Q^solv is strictly smaller than Q, the algebraic closure of the rationals, the Algebraic numbers, and in particular the roots of most equations with degree bigger than 4 are not in Q^solv. Galois theory identifies extensions with solvable Galois group with ones definable by (iterated) radicals.John Z (talk) 21:00, 8 September 2012 (UTC)[reply]

John, yes, that is what I meant. I'm a bit out of my depth here, but it sounds like you're saying that a cubic with 3 real roots is expressible using radicals (by which we mean here real radicals of real numbers, starting with integers), which I was under the impression is impossible in general. I suspect that solvability is critically dependent on the field (R or C). For example, x² + 1 is an irreducible element over R, but is factorizable over C, and related subtleties may affect the interpretation of Galois theory. Or something like that – as I said, this is a bit beyond me. — Quondum 23:20, 8 September 2012 (UTC)[reply]

No, you are right, you can't always get all the roots of a cubic with three real roots by only using real radicals of real numbers. You may have to take roots of complex numbers, but that is OK since Q[i] for instance is an extension with a solvable (abelian, even cyclic) galois group; you get it by taking radicals. When we have enough roots of unity in the base, cyclic extensions - ones with cyclic galois group - correspond to taking an nth root of something in the base. Solvable extensions, ones with solvable galois groups correspond to taking iterated radicals. The maximal solvable extension is built up by taking all the iterated radicals, as RJFJR wants. Take all the roots of stuff in Q. Adjoin them to Q & you get some field. Then take all the roots of stuff in there, and go on ad infinitum.

So to get the solution of such a cubic, you can think of taking an extension adjoining i, or the square root of some negative number (given in the cubic formula) & then take some cube root of something in that extension to get the solution of some cubic using the cubic formula. The imaginary parts can cancel out and you land back in R, but to write the solution using radicals, taking them in fields not embeddable in R is necessary.

Another interesting & even more important, probably, subfield of Q^solv is Q^ab, the maximal abelian extension of Q, the biggest extension of Q with abelian galois group. It is a real, non-obvious theorem, the Kronecker–Weber theorem, that this is the same field as that gotten by adjoining all roots of 1, the same field as taking all Cyclotomic fields together, what our Class field theory calls

\mathbb {Q} (\mu _{\infty })

.John Z (talk) 00:05, 9 September 2012 (UTC)[reply]

Oops, I see where you are coming from, Quondum. I misread the question, as if RJFJR was asking for the name of the field gotten from taking radicals, then "radicals of previously obtained numbers" etc., as are used in the cubic & quartic formula. But he says "radicals of integers" only, which eliminates many fields generated by solutions of cubics & quartics using radicals. Hmm, not sure of a name for that, though have an idea I need to think about more, with less beer. Googling finds a sloppy statement in a reference work & a book with an incorrect definition that mistakenly says what you (reasonably) thought I was saying, that "solvable by radicals" is the same as "solvable by radicals of rationals/integers", ignoring the dependence on the base field you rightly observed, and thus states an incorrect theorem that implies there is a cubic formula using only radicals of rational numbers! To err is human, but the works of Cardano, Tartaglia, Abel, Galois, Artin etc are divine.John Z (talk) 08:01, 9 September 2012 (UTC)[reply]

I didn't notice the "integer" part either. For a while there was an article called radical integer that might have provided an answer to this, but it was AfD'd (disclosure: I filed the AfD) on the grounds that the term was not standard and the claimed result was not reliably sourced. Anyway the claimed theorem, if I have it right, is that if a number is obtainable by radicals starting with the rationals, and is also an algebraic integer, then it's obtainable by radicals starting with the integers. Rich Schroeppel coined the term radical integer for (I think) the latter concept.

It would be very nice to know if this result has been published and confirmed. I would be happy to have the content back, once it's no longer sourced to Mathworld or Eric Weisstein. --Trovatore (talk) 08:22, 9 September 2012 (UTC)[reply]

Interesting AfD. From it, "Radical number" or "Radical integer" seems to refer to the iterated radicals, solvable field extension case. Although "radical number" in Maple seems to be more at what RJFJR is getting at: Radicals of integers along with i (but what about other roots of unity?). The ref for the article is both confused & confusing, and I think the correct decision was taken. The claimed theorem sounds plausible, but it is hard to be sure we're even getting the statement right.

The only thing I think I can observe about the original question is that this field of radical numbers sits in between two fields with accepted names. If you call it Q^rad, then Q < Q^ab < Q^rad < (Q^ab)^ab < Q^solv. The galois groups of a radical extension of Q (radicals of integers) with enough roots of 1 added to make it galois are metabelian - solvable, with derived length two, group extensions of abelian by abelian. By Kronecker-Weber, the structure of finite abelian groups and galois theory, a partial converse is that an extension of Q with metabelian galois group comes from cyclic, radical extensions of a cyclotomic field, where we take radicals of the base, cyclotomic field, however. (Q^ab)^ab is the "maximal metabelian extension of Q" - the biggest extension of Q with metabelian galois group, and is the maximal abelian extension of Q^ab, and comes from taking radicals of all elements of Q^ab = $\mathbb {Q} (\mu _{\infty })$ . However, the Casus irreducibilis, the necessity of using radicals of complex numbers to solve cubic (& higher) equations, that Quondum noted, (I didn't notice our article on it before) appears to show that Q^rad, while being an extension of Q^ab is strictly smaller than (Q^ab)^ab.John Z (talk) 08:50, 11 September 2012 (UTC)[reply]

September 8

"excessive" shuffling

One of the regulars in my weekly bridge game complains that I shuffle the deck too many times, leading to perverse distributions. Should I take it seriously? —Tamfang (talk) 05:53, 8 September 2012 (UTC)[reply]

Well, I suppose certain shuffling patterns might be self-reversing. Let's say you just cut the deck in half, then put the top half on the bottom. The next time you do that, you should reverse the cut, if you manage to cut it in exactly the same spot. However, the odds of you having such a self-reversing shuffle seem quite low. Otherwise, if your shuffle truly randomizes the cards, then repeating it shouldn't make them any less random, on average. StuRat (talk) 06:03, 8 September 2012 (UTC)[reply]

Disclaimer: What I'm about to relate was learned by word of mouth, so may be apocryphal. When bridge software was first being developed, the programmers initially thought they had a bug with their shuffling algorithm, because they kept getting very unusual distributions. So they went back and picked over the code, but there was no problem with it. It turns out that when shuffling by hand, most people do a very poor job of randomizing the deck, and so the distributions seemed odd because the programmers were all used to insufficiently randomized decks. So it's possible that your shuffling is leading to unusual distributions, but the fault lies with everyone else, not with you.--121.73.35.181 (talk) 06:34, 8 September 2012 (UTC)[reply]

Well a poorly shuffled deck is more likely to give interesting sequences in the cards and a more entertaining game. --Salix (talk): 09:20, 8 September 2012 (UTC)[reply]

Actually I think a poorly shuffled deck tends to give less distributional hands. You probably don't notice it except in the tails, which means you probably don't notice it — maybe you don't get as many nine-card suits, but then, how many do you expect?

(In the extreme case, say you don't shuffle at all. Then any trick in which everyone followed suit, results in one card of that suit to each player.) --Trovatore (talk) 10:26, 8 September 2012 (UTC)[reply]

Some expert bridge players can anecdotally do a perfect shuffle, which after a number of such shuffles (8, I think it was) I've heard can return the deck to its original state. Under the assumption of adequately unpredictable shuffling, the claim can be ignored as the distribution will become more random with each shuffle. Under the assumption of chosen shuffles (are you a card sharp?), the number of shuffles hardly matters: the shuffler can in principle predict (and control) the result. — Quondum 06:46, 8 September 2012 (UTC)[reply]

Here is a source for what 121.73.. wrote. Seven shuffles should be enough to mix a deck of cards thoroughly. Players may not like it because of odd distributions, or they may have another reason to prefer less shuffling ...

"... These players had figured out that the cards were not being randomly shuffled, and that they could predict the distributions of cards by knowing what the deck looked like at the end of the previous hand." Ssscienccce (talk) 00:21, 9 September 2012 (UTC)[reply]

This is really interesting — led me to the Persi Diaconis article and his "seven shuffles" result. It says that after seven shuffles, the total variation distance falls below 0.5 and then is halved every shuffle. If I've correctly understood what that means (which I'm not certain), then in order to be sure you had about the right probability for a nine-card suit, you'd have to shuffle roughly 18 times.

According to our bridge probabilities article, the probability of a nine-card suit is 0.00037, which means 2700-to-1 against. I think I've had about three nine-card suits in my life, which you'd expect in about 5000 hands. I think I've played more than that but I'm not sure (it's been some time since I've played at all); anyway I can't claim that the number is too high or too low with any good statistical significance. --Trovatore (talk) 08:44, 9 September 2012 (UTC)[reply]

It could even be that the effect this player is noticing is that your shuffling is actually more random than other people's in the group. It's possible that by shuffling too few times, the suits get distributed more evenly than random (since they'll tend to be clumped after a hand and then spread out at regular intervals). These distributions might feel more random because there's less variance in hand strength, but aren't. I've heard of people complaining about the shuffling in online games (which should be much closer to random than human shuffling), because they have a false conception of what random should look like. That said, most likely everyone's shuffling is fine and it's just confirmation bias on the part of this other player. Rckrone (talk) 01:15, 9 September 2012 (UTC)[reply]

Sorry, didn't read carefully that 121.73.35.181 already made the same point. Rckrone (talk) 15:18, 9 September 2012 (UTC)[reply]

Tesseract

i am looking for information to help me better understand the tesseract and the 4D. — Preceding unsigned comment added by 205.142.178.36 (talk) 19:59, 8 September 2012 (UTC)[reply]

See the article tesseract, it's great. You can also read Flatland by Abbott and watch Carl Sagan's Cosmos episode 10, "The Edge of Forever." available on line. μηδείς (talk) 20:09, 8 September 2012 (UTC)[reply]

As long as we're referencing works of fiction, check out —And He Built a Crooked House by Heinlein. In addition to being a very entertaining (well, at least entertainingly written) story in its own right, the action takes place in rooms comprising the (hyper)faces of a tesseract, and the description of how they are connected is meticulously accurate (I worked it out). (The opening passage about Americans, Californians, Angelenos, Hollywood residents, Laurel Canyon, Lookout Mountain Avenue, and finally the protagonist, is worth memorizing.) --Trovatore (talk) 21:41, 8 September 2012 (UTC)[reply]

Possibly of interest, one of my favorite Youtube movies shows the 120-cell (which is to the tesseract as the dodecahedron is to the cube) as a tiling of spherical 3-space. —Tamfang (talk) 04:37, 9 September 2012 (UTC)[reply]

Star Trek Special Effect

At time signature 18:12 on this Star Trek episode http://www.youtube.com/watch?v=hornvnjML88 you see the classic yellow-and black rotating interference band from the vintage Star Trek by Spock's head. I know it is caused by two interfering rotating patterns. What are those underlying patterns? Thanks. μηδείς (talk) 20:06, 8 September 2012 (UTC)[reply]

I think it's simply two offset radial patterns, as seen here. -- BenRG (talk) 04:07, 9 September 2012 (UTC)

[reply]

Exactly! Wonderful. I had been guessing two spirals, which I think would have been way more checkered. Great! μηδείς (talk) 04:21, 9 September 2012 (UTC)[reply]

Resolved

Can you give the url of the page in which that image was embedded? μηδείς (talk) 04:49, 9 September 2012 (UTC)[reply]

I updated the original link. It's an interesting web site and I hadn't even noticed since I just found the image in Google Image Search. -- BenRG (talk) 05:20, 9 September 2012 (UTC)[reply]

Extrapolating sums of powers

Let's say we define G(N, P) = {1^P + ... + N^P}. Now given that A = G(N, 1), B = G(N, 2), and C = G(N, 3) are easy to calculate, is there any way to extrapolate to get the value of D = (G(N, N) mod (N + 1)) using A, B, and/or C (by inspecting the differences between their respective values, or what have you)? If not, can we, at the very least, infer anything whatsoever about D?

76.25.96.107 (talk) 22:08, 8 September 2012 (UTC)[reply]

See Faulhaber's formula Dmcq (talk) 22:55, 8 September 2012 (UTC)[reply]

Sorry, I should have been clearer: I'm seeking an efficient method for any given N.

76.25.96.107 (talk) 23:04, 8 September 2012 (UTC)[reply]

This only works for a subset of N, but is very efficient for that subset (unless I've bungled it): if N+1 is a prime number, then D = N. This follows from Fermat's little theorem. — Quondum 23:32, 8 September 2012 (UTC)[reply]

An efficient method is to calculate the first few values and then look up the Encyclopaedia of Integer Sequences, but that is only for a limited number of values ;-) Sticking 1,2,0,4,3,6,0,6 into that gives A204187. Unfortunately except for the case of N+1 a prime the main reference seems to be 'R. K. Guy, Unsolved Problems in Number Theory, A17' which indicates anything simple may be a bit much to ask. Always worth a good look though if it has come up in some different context. Dmcq (talk) 08:56, 9 September 2012 (UTC)[reply]

According to the OEIS page, we can expand the "efficient" cases:

If N+1 is prime, D = N
If N+1 is a multiple of 4, D = 0
Otherwise, there appears to be no known efficient way to calculate D.

— Quondum 09:38, 9 September 2012

Well it looks to me like there probably are some rules depending on the factorization of N+1. For instance for the table of values it looks like the value for p^2 is p(p-1) and for 2.(2n+1) is 2n+1 and for 3.p is 2.p. Though how one gets 52 from 65, I don't know - one would expect both factors 5 and 13 to enter if one does. Dmcq (talk) 15:43, 9 September 2012 (UTC)[reply]

N + 1 = p₁p₂ ⇒ D = p₁(p₂ – 1)? Err... It is looking vaguely reminiscent of the Euler totient function, with some unpredictable deviations. If N is quite small (< 2¹⁶ say), and one can afford a few milliseconds to calculate each D, I'd just do it on a computer using modulo arithmetic. — Quondum 20:32, 9 September 2012 (UTC)[reply]

Hmmmm, can't this be simplified by using that this is

{\frac {B_{n+1}(n+1)-B_{n+1}(0)}{n+1}}{\text{mod }}(n+1)

The terms in the Bernoulli polynomial evaluated at n+1 are mostly divisible by n+1 and then also zero when reduced mod n+1, so it seems to me that you don't have many nonzero terms left... Count Iblis (talk) 21:40, 9 September 2012 (UTC)[reply]

How could that possibly lead to a simplification though - the Bernoulli numbers are notoriously inefficient to calculate, aren't they?

66.87.126.39 (talk) 03:35, 10 September 2012 (UTC)[reply]

You end up with one Bernoulli number. You can take Modulo n+1 in the entire polynomial, if the denominator is not a zero-divisor, you obviously get zero, if not then you still have a high enough power of n+1 left to make the term zero. This is true for all terms except the first term after the contant term. That term is proportional to a Bernoulli number and you need to evaluate this Modulo (n+1) (with the right convention if the re are zero divisors). I have to look into this in more detail, but it seems to me that the Von Staudt–Clausen theorem can be used to simply this. Count Iblis (talk) 04:28, 10 September 2012 (UTC)[reply]

So, we need to calculate the coefficient of the lowest nonzero power of x in $B_{n+1}(x)$ , which for even N is the coefficient of x, and for odd N is the coefficient of x^2. Using the recursion relation:

{\frac {B_{n}(x)}{dx}}=nB_{n-1}(x)

you find that for even n this is

c_{\text{even}}=(n+1)B_{n}

and for odd n this is

c_{\text{odd}}=n{\frac {(n+1)}{2}}B_{n-1}

For even n the summation is

c_{\text{even}}{\text{ mod}}(n+1)=(n+1)B_{n}{\text{ mod}}(n+1)

because multiplying with x and setting x = n+1 and then dividing by n+1 yields a factor 1. In case of odd n, the sumation is

(n+1)c_{\text{odd}}{\text{ mod}}(n+1)=n{\frac {(n+1)^{2}}{2}}B_{n-1}{\text{ mod}}(n+1)

because now the term must be multiplied by x^2 and divided by n+1 and setting x = n+1 yields a factor of n+1.

In both cases, the Bernoulli numbers can be reduced Modulo 1, as that shifts the terms by a number proportional to n+1 which is then zero as we're computing modulo n+1. The Von Staudt–Clausen theorem then yields for even n the expression

-\sum _{p-1|n}(n+1)p^{-1}{\text{ mod}}(n+1)

where the p in the summation are prime numbers. This has to be evaluated with the convention that P^(-1) multiples (n+1) before we reduce modulo (n+1). If p is not a divisor of n+1, this is the same as computing the inverse of p modulo n+1 first, but then we see that term becomes zero. So, the summation is equal to the above expression where we now only sum over the prime numbers that divide n+1.

For odd n, the summation can be written as

-\sum _{p-1|n-1}n{\frac {(n+1)^{2}}{2}}p^{-1}{\text{ mod}}(n+1)

Then as in the even case, primes that don't divide n+1 yield zero. But now also odd primes that divide n+1 yield zero, because you can then let p divide (n+1)/2 and the term will still be proportional to n+1 = 0 Mod(n+1). So, the only nonzero contribution can come from p = 2. If n+1 is divisible by 4, it will remain zero as p = 2 will divide (n+1)/2. Only if n + 1 is not divisible by 4, so when n = 4 k + 1 will the term be nonzero, it then becomes

{\frac {n+1}{2}}

To see this, you can use the fact that n = -1 mod(n+1) and (n+1)^2/4 = (n+1)/2 (n+1)/2 = (n+1)/2 [(n+1)/2 - 1 + 1] = (n+1)/2 mod (n+1) because (n+1)/2 - 1 is an even number. Count Iblis (talk) 16:25, 11 September 2012 (UTC)[reply]

Cube: 2 Hypercube

I do not want to be a bother to Mathmatics but i was wondering how accurate is the movie Cube 2 Hypercube, i know its all about tesseracts and 4D? Can any one explain how it works to me? — Preceding unsigned comment added by 205.142.178.36 (talk) 23:34, 8 September 2012 (UTC)[reply]

|

A 3D projection of an 8-cell performing a double rotation about two orthogonal planes

Did you not find the above answers to your question helpful? μηδείς (talk) 00:29, 9 September 2012 (UTC)[reply]

No this is about the movie in general — Preceding unsigned comment added by 76.16.47.115 (talk) 02:46, 9 September 2012 (UTC)[reply]

It's a horror movie. You might do well to ask a specific mathematical question here, or a plot question on the entertainment desk. μηδείς (talk) 04:22, 9 September 2012 (UTC)[reply]

Explain how it would work in math — Preceding unsigned comment added by 76.16.47.115 (talk) 05:00, 9 September 2012 (UTC)[reply]

The surface of a cube consists of six squares, each of which is connnected via its four sides to four of the other five square that make up the surface of the cube. In the same way, the "hypersurface" of a tessaract consists of eight cubes, each of which is connected via its six faces to six of the other seven cubes that make up the "hypersurface" of the tessaract.

If you take a tour of the squares around a cube, always going straight on i.e. exiting each square on the opposite side that you entered it, then you go through four different squares before returning to where you started. In the same way, if you take a tour of the cubes around a tesseract, always going straight on i.e. exiting each cube on the opposite face that you entered it, then you go through four different cubes before returning to where you started. Gandalf61 (talk) 08:15, 9 September 2012 (UTC)[reply]

You really should read the tesseract article, read Flatland, and watch the Edge of Forever episode available all over the internet if you want to understand the fourth dimension. I would start with the cosmos episode: here at google. μηδείς (talk) 21:39, 9 September 2012 (UTC)[reply]

September 9

Isomorphism between G/N x N ~= G

Is there always an isomorphism between the quotient of a group G and a normal subgroup N with the direct product of N to G?

G/N × N ≈ G

Is there a theorem for that? --helohe (talk) 12:39, 9 September 2012 (UTC)[reply]

No, because it's not true. The simplest counterexample is

G=S_{3}

, which contains normal a subgroup generated by the 3-cycles,

N=\langle (123)\rangle

. Then

G/N\cong \mathbb {Z} _{2}

, but

G

is not abelian, so

G\ncong N\times \mathbb {Z} _{2}

. — Preceding unsigned comment added by COVIZAPIBETEFOKY (talk • contribs) 13:16, 9 September 2012 (UTC)[reply]

I think it should work if you use the semidirect product instead. —Kusma (t·c) 08:35, 10 September 2012 (UTC)[reply]

This fails even for abelian groups: C₄/C₂ ≃ C₂, but C₄ is not isomorphic to

C_{2}\rtimes C_{2}

. In general, G is the semidirect product of N and G/N if and only if the exact sequence

1\longrightarrow N\longrightarrow G\longrightarrow G/N\longrightarrow 1

is right split.—Emil J. 15:25, 10 September 2012 (UTC)[reply]

Thank you for the correction. —Kusma (t·c) 07:11, 11 September 2012 (UTC)[reply]

Finding the constituent numbers in a list that add up to partial totals

I was wondering of there was a methodology I could use for future reference for a problem I had the other day. I had 28 undated invoices, each with amounts and I had three checks that I knew were payments, each of more than one invoice on that list but not which ones and they were not in any order and I had to match the payments to which invoices they covered. Complicating matters, I also knew that the checks were only partial payment of the total invoices--they would match exactly some combination of the invoices but not all. I took me a few hours to actually figure it out by trial and error, playing with the numbers in excel, and I know there has got to be a better way. Let me provide the actual numbers and the solution so you'll know what I mean (I also think there might be a computer program that might solve this but I had no idea how to look for such a program). Here goes:

Invoices

Template:MultiCol $18.43
$23.91
$7.32
$234.65
$10.13
$11.55
$34.60

| class="col-break " | $67.37
42.81
$75.44
$233.45
$98.43
$91.36
$31.04

| class="col-break " | $72.94
$58.38
$26.48
$95.02
$78.45
$66.57
$48.30

| class="col-break " | $57.67
$49.30
$23.86
$87.88
$89.47
$35.33
88.40 Template:EndMultiCol

Checks

check 1: $327.79
check 2: $437.66
check 3: $381.18

As you can see, I've color coded the solution. The red invoice amounts equal the red check exactly, and the same for green and blue and the uncolored were those invoices that were not paid. But I started with no idea which invoices equaled which checks. I never want to have to do this again blindly.--108.54.25.10 (talk) 13:35, 9 September 2012 (UTC)[reply]

The most obvious answer is to keep better track of things in the first place (date your invoices), so you don't need to try to figure it out after. However, a trial-and-error program is the obvious solution. If you want, I could even write you one and send it to you. However, there is the possibility of multiple solutions. StuRat (talk) 14:53, 9 September 2012 (UTC)[reply]

This is the Subset sum problem. Taemyr (talk) 15:02, 9 September 2012 (UTC)[reply]

Looking at the most brute force approach of just trying every amount in one of 4 categories (including those that don't match anything), I get an unmanageable 4^28 possible combos, or some 72 quadrillion. That approach is definitely out. A more elegant approach might be to just look at which items add up to the lowest total first. StuRat (talk) 15:17, 9 September 2012 (UTC)[reply]

Interesting. Do you know of an existing program online to plug in numbers and get a solution? If not, I will take you up on that offer Stu (I use macs, which I thought you might need to know to make the program work for me) My email is ammtss AT yahoo DOT com. By the way, they are not my invoices. I am an attorney matching proof to make a coherent presentation to a jury on a property damage suit; the invoices are for building supplies paid to a third party by my client for the rebuild after massive flood damage. There were some clues in the invoices that helped me narrow down into categories and mix and match from there or I don't think I could have done it.--108.54.25.10 (talk) 15:31, 9 September 2012 (UTC)[reply]

That explains it then. I was wondering how anyone could solve such a complex problem by hand. StuRat (talk) 16:50, 9 September 2012 (UTC)[reply]

I have some bad news for you. When I said there might be multiple solutions, that was apparently an understatement. I wrote a program to solve it, and came up with 623 possible solutions. Your solution was the 11th. Here's just one more (the 63rd):

  78.45
  58.38               91.36
  57.67               89.47
  48.30               75.44
  35.33               72.94
  23.91    233.45     42.81
  18.43     98.43     34.60
+  7.32   + 49.30   + 31.04
=======   =======   =======
 327.79    381.18    437.66

So, the solution you found and presented in court may not be correct. Therefore, even with a program, you will end up with a large number of possibilities, and no way to know which one is correct. I'm afraid you're down to either presenting the evidence as your client gave it to you, getting your client to give you better info, or lying in court and claiming that one of the solutions you found is the correct one, when you really don't know that. StuRat (talk) 05:51, 10 September 2012 (UTC)[reply]

What strikes me about the figures is they seem rather peculiar. There's no duplicates and they don't look like the prices people charge and don't evem look like ones with some tax added. The first digits don't follow Benford's law at all. I've had unfortunately soemtimmes had to deal with money which hadn't been properly tracked and there's all sorts of things one can check - the people who wrote a checque, what kind of item the amounts correspond to, the dates etc. In this case your client can probably ietmize most of the things bought and when and a check with the suppliers catalogue could then say for the ones that were remembered what the price was. The dates of the remainder with a list of where the others fit would then prompt more and might suggest what was being worked on at the time. Dmcq (talk) 16:40, 10 September 2012 (UTC)[reply]

I would never try to brute-force problems like this, and you don't have to. First of all, you don't need to check 4^28 combinations, you can split the problem and try to cover the smallest check first.

That would be 2^28 combinations if brute-forced, but you can sort the items by size and take the biggest item that will "fit." Let's say you want to cover the $327.79 check first. The $233.45 item will fit so it'll reduce the problem to another one, namely, "Try to cover the remaining $94.34 using the remaining 27 items." The $98.43 item won't fit but the $91.36 item will - that'll be layer 3, and you'll immediately see that there is no solution to the remaining $2.98. So you backtrack to recursion layer 2 and try the next item on the list, $89.47. etc, etc.

The point is that you can eliminate large classes of solutions quickly, for example that neither the combination 233.45+91.36 nor 233.45+89.47 is part of the $327.79 check.

And what happens to the other items can be determined if/when you have all solutions to the $327.79 anyway. Any solution will eliminate some of the 28 items, so by solving the smallest check first, you'll even save quite a bit of work later on.

Maybe some backtracking of the invoices or the checks can narrow the possibilities a bit. I doubt that it can kill 622 of your 623 solutions though. The quantities are still odd. They look close to equidistributed between $0 and $100, except that items below $5 are multiplied by 100; this looks like a made-up math problem rather than actual invoices.

Kudos to Dmcq, you beat me to Benford. - ¡Ouch! (^{hurt me} / _{more pain}) 07:07, 11 September 2012 (UTC)[reply]

The approach you list is close to what I did in my program, although I started with the smallest item, not the largest. I'm amazed at how fast the program is (about 4 seconds of real time on my low-end PC, increasing to 53 seconds if I take the time to print out the results), considering how utterly unworkable the full brute force approach is. StuRat (talk) 16:58, 11 September 2012 (UTC)[reply]

Benford's law can only be applied to data that are distributed across multiple orders of magnitude. Ssscienccce (talk) 17:57, 11 September 2012 (UTC)[reply]

Here's the list of all 623 solutions:

Full solution set.

1 :

       95.02
       58.38
       57.67               88.40
       42.81               87.88
       26.48               78.45
       18.43               75.44
       11.55    233.45     48.30
       10.13     98.43     35.33
     +  7.32   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

2 :

      233.45     98.43    234.65
       34.60     88.40     75.44
       23.86     78.45     49.30
       18.43     58.38     42.81
       10.13     31.04     23.91
     +  7.32   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

3 :

                           88.40
      233.45               87.88
       34.60     98.43     67.37
       23.86     91.36     66.57
       18.43     89.47     49.30
       10.13     75.44     42.81
     +  7.32   + 26.48   + 35.33
     =======   =======   =======
      327.79    381.18    437.66

4 :

                           91.36
                           89.47
      233.45     87.88     72.94
       34.60     75.44     57.67
       23.86     67.37     48.30
       18.43     58.38     35.33
       10.13     49.30     31.04
     +  7.32   + 42.81   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

5 :

                           98.43
      233.45     87.88     95.02
       34.60     75.44     91.36
       23.86     67.37     57.67
       18.43     58.38     48.30
       10.13     49.30     35.33
     +  7.32   + 42.81   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

6 :

       67.37
       58.38
       57.67               91.36
       49.30               89.47
       35.33               75.44
       23.86    233.45     72.94
       18.43     87.88     42.81
       10.13     48.30     34.60
     +  7.32   + 11.55   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

7 :

       67.37
       58.38
       57.67
       49.30               98.43
       35.33               95.02
       23.86    233.45     91.36
       18.43     87.88     75.44
       10.13     48.30     42.81
     +  7.32   + 11.55   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

8 :

       95.02
       88.40               98.43
       49.30               78.45
       35.33               75.44
       23.86    233.45     58.38
       18.43     87.88     57.67
       10.13     48.30     42.81
     +  7.32   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

9 :

       95.02
       88.40               91.36
       49.30               89.47
       35.33               75.44
       23.86    233.45     72.94
       18.43     87.88     42.81
       10.13     48.30     34.60
     +  7.32   + 11.55   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

10 :

       95.02
       75.44               89.47
       49.30               88.40
       42.81               72.94
       23.91    233.45     67.37
       23.86     87.88     57.67
       10.13     48.30     35.33
     +  7.32   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

11 :

       95.02
       67.37               89.47
       66.57               88.40
       31.04               75.44
       26.48    233.45     72.94
       23.86     87.88     58.38
       10.13     48.30     34.60
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

12 :

                           89.47
       95.02               78.45
       67.37               58.38
       66.57               57.67
       31.04               48.30
       26.48               35.33
       23.86    233.45     34.60
       10.13     98.43     23.91
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

13 :

       95.02               89.47
       67.37               75.44
       66.57               72.94
       31.04               58.38
       26.48               57.67
       23.86    233.45     48.30
       10.13     98.43     23.91
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

14 :

       95.02
       67.37               91.36
       66.57               87.88
       31.04               78.45
       26.48               75.44
       23.86    233.45     58.38
       10.13     98.43     34.60
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

15 :

       95.02               89.47
       67.37               88.40
       66.57               75.44
       31.04               58.38
       26.48               48.30
       23.86    233.45     35.33
       10.13     98.43     23.91
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

16 :

       95.02
       67.37               89.47
       66.57               88.40
       31.04               75.44
       26.48               72.94
       23.86    233.45     58.38
       10.13     98.43     34.60
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

17 :

       95.02               89.47
       67.37               75.44
       66.57               72.94
       31.04               57.67
       26.48               48.30
       23.86    233.45     35.33
       10.13     98.43     34.60
     +  7.32   + 49.30   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

18 :

       91.36               89.47
       78.45               88.40
       67.37               75.44
       49.30    233.45     72.94
       23.86     87.88     58.38
       10.13     48.30     34.60
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

19 :

       89.47
       72.94               95.02
       58.38               88.40
       34.60               75.44
       31.04    233.45     67.37
       23.91     87.88     57.67
       10.13     48.30     35.33
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

20 :

       89.47
       72.94
       58.38     98.43
       34.60     88.40    233.45
       31.04     75.44     78.45
       23.91     57.67     66.57
       10.13     42.81     35.33
     +  7.32   + 18.43   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

21 :

       89.47
       72.94               95.02
       58.38               88.40
       34.60               75.44
       31.04               67.37
       23.91    233.45     57.67
       10.13     98.43     35.33
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

22 :

       89.47
       72.94               88.40
       58.38               87.88
       34.60               78.45
       31.04               75.44
       23.91    233.45     48.30
       10.13     98.43     35.33
     +  7.32   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

23 :

       95.02
       67.37               89.47
       57.67               88.40
       35.33               75.44
       31.04    233.45     72.94
       23.91     87.88     58.38
       10.13     48.30     34.60
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

24 :

       95.02               91.36
       67.37               88.40
       57.67               78.45
       35.33               66.57
       31.04               48.30
       23.91    233.45     34.60
       10.13     98.43     18.43
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

25 :

                           89.47
       95.02               78.45
       67.37               66.57
       57.67               58.38
       35.33               48.30
       31.04               34.60
       23.91    233.45     26.48
       10.13     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

26 :

       95.02
       67.37               91.36
       57.67               87.88
       35.33               78.45
       31.04               75.44
       23.91    233.45     58.38
       10.13     98.43     34.60
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

27 :

       95.02
       67.37               89.47
       57.67               88.40
       35.33               75.44
       31.04               72.94
       23.91    233.45     58.38
       10.13     98.43     34.60
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

28 :

       95.02               89.47
       67.37               75.44
       57.67               72.94
       35.33               66.57
       31.04               48.30
       23.91    233.45     34.60
       10.13     98.43     26.48
     +  7.32   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

29 :

       98.43     89.47
       95.02     87.88
       58.38     75.44    233.45
       34.60     67.37     78.45
       23.91     31.04     66.57
       10.13     18.43     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

30 :

                 89.47
       98.43     67.37
       95.02     57.67
       58.38     49.30    233.45
       34.60     48.30     78.45
       23.91     31.04     66.57
       10.13     26.48     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

31 :

                           91.36
       98.43               88.40
       95.02               72.94
       58.38               49.30
       34.60    233.45     42.81
       23.91     87.88     35.33
       10.13     48.30     31.04
     +  7.32   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

32 :

       98.43               89.47
       95.02               88.40
       58.38               72.94
       34.60    233.45     67.37
       23.91     87.88     57.67
       10.13     48.30     35.33
     +  7.32   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

33 :

       75.44
       67.37               91.36
       49.30     98.43     87.88
       48.30     88.40     72.94
       35.33     78.45     66.57
       34.60     58.38     57.67
       10.13     31.04     42.81
     +  7.32   + 26.48   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

34 :

       95.02     91.36     89.47
       87.88     78.45     88.40
       49.30     67.37     75.44
       42.81     57.67     72.94
       35.33     48.30     58.38
       10.13     26.48     34.60
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

35 :

       95.02
       57.67
       48.30               88.40
       34.60               87.88
       31.04     98.43     67.37
       23.86     91.36     66.57
       18.43     89.47     49.30
       11.55     75.44     42.81
     +  7.32   + 26.48   + 35.33
     =======   =======   =======
      327.79    381.18    437.66

36 :

       89.47
       88.40
       57.67     87.88
       31.04     75.44    233.45
       23.91     67.37     78.45
       18.43     58.38     66.57
       11.55     49.30     35.33
     +  7.32   + 42.81   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

37 :

       89.47               95.02
       88.40               75.44
       57.67               67.37
       31.04               58.38
       23.91               48.30
       18.43    233.45     42.81
       11.55     98.43     26.48
     +  7.32   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

38 :

       87.88
       72.94
       42.81               95.02
       31.04               88.40
       26.48               75.44
       23.91               67.37
       23.86    233.45     57.67
       11.55     98.43     35.33
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

39 :

       91.36
       87.88               95.02
       48.30     98.43     89.47
       31.04     88.40     67.37
       26.48     75.44     66.57
       23.86     57.67     49.30
       11.55     42.81     35.33
     +  7.32   + 18.43   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

40 :

       91.36
       87.88               95.02
       48.30     98.43     88.40
       31.04     89.47     75.44
       26.48     66.57     67.37
       23.86     49.30     57.67
       11.55     42.81     35.33
     +  7.32   + 34.60   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

41 :

       91.36
       87.88               89.47
       48.30               88.40
       31.04               75.44
       26.48               72.94
       23.86    233.45     58.38
       11.55     98.43     34.60
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

42 :

       91.36
       87.88               95.02
       48.30               88.40
       31.04               75.44
       26.48               67.37
       23.86    233.45     57.67
       11.55     98.43     35.33
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

43 :

       87.88
       72.94     98.43    233.45
       66.57     88.40     67.37
       57.67     78.45     49.30
       23.86     58.38     42.81
       11.55     31.04     34.60
     +  7.32   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

44 :

       87.88
       72.94              233.45
       66.57     98.43     67.37
       57.67     91.36     49.30
       23.86     89.47     42.81
       11.55     75.44     34.60
     +  7.32   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

45 :

                           88.40
                           78.45
       87.88               67.37
       72.94               58.38
       66.57               42.81
       57.67               34.60
       23.86    233.45     31.04
       11.55     98.43     26.48
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

46 :

                           91.36
       87.88               89.47
       72.94               75.44
       66.57               67.37
       57.67               42.81
       23.86    233.45     34.60
       11.55     98.43     26.48
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

47 :

                           89.47
       87.88               88.40
       72.94               75.44
       66.57               58.38
       57.67               48.30
       23.86    233.45     35.33
       11.55     98.43     23.91
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

48 :

                           91.36
       87.88               89.47
       72.94               75.44
       66.57               48.30
       57.67               42.81
       23.86    233.45     35.33
       11.55     98.43     31.04
     +  7.32   + 49.30   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

49 :

       78.45               89.47
       66.57               87.88
       58.38               72.94
       48.30               67.37
       26.48               42.81
       23.86    233.45     34.60
       18.43     98.43     31.04
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

50 :

       78.45
       66.57               91.36
       58.38               89.47
       48.30               75.44
       26.48               72.94
       23.86    233.45     42.81
       18.43     98.43     34.60
     +  7.32   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

51 :

                 89.47
       88.40     67.37
       87.88     57.67     98.43
       66.57     49.30     95.02
       35.33     48.30     91.36
       23.86     31.04     75.44
       18.43     26.48     42.81
     +  7.32   + 11.55   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

52 :

       88.40
       87.88              233.45
       66.57     98.43     67.37
       35.33     91.36     49.30
       23.86     89.47     42.81
       18.43     75.44     34.60
     +  7.32   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

53 :

                           95.02
                           67.37
       88.40               57.67
       87.88               49.30
       66.57     98.43     48.30
       35.33     91.36     42.81
       23.86     89.47     34.60
       18.43     75.44     31.04
     +  7.32   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

54 :

       88.40
       87.88     98.43    234.65
       66.57     91.36     75.44
       35.33     67.37     49.30
       23.86     58.38     42.81
       18.43     34.60     23.91
     +  7.32   + 31.04   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

55 :

                           91.36
       88.40               89.47
       87.88               75.44
       66.57               67.37
       35.33               42.81
       23.86    233.45     34.60
       18.43     98.43     26.48
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

56 :

                           89.47
       88.40               75.44
       87.88               72.94
       66.57               58.38
       35.33               57.67
       23.86    233.45     48.30
       18.43     98.43     23.91
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

57 :

                           91.36
       88.40               89.47
       87.88               78.45
       66.57               58.38
       35.33               42.81
       23.86    233.45     34.60
       18.43     98.43     31.04
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

58 :

       88.40               91.36
       87.88               89.47
       66.57               75.44
       35.33               72.94
       23.86    233.45     42.81
       18.43     98.43     34.60
     +  7.32   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

59 :

       78.45
       58.38
       57.67              233.45
       48.30     98.43     67.37
       35.33     91.36     49.30
       23.91     89.47     42.81
       18.43     75.44     34.60
     +  7.32   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

60 :

       78.45               91.36
       58.38               89.47
       57.67               75.44
       48.30               67.37
       35.33               42.81
       23.91    233.45     34.60
       18.43     98.43     26.48
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

61 :

       78.45
       58.38
       57.67              234.65
       48.30               67.37
       35.33               66.57
       23.91    233.45     31.04
       18.43     98.43     26.48
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

62 :

       78.45               89.47
       58.38               87.88
       57.67               72.94
       48.30               67.37
       35.33               42.81
       23.91    233.45     34.60
       18.43     98.43     31.04
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

63 :

       78.45
       58.38               91.36
       57.67               89.47
       48.30               75.44
       35.33               72.94
       23.91    233.45     42.81
       18.43     98.43     34.60
     +  7.32   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

64 :

                           98.43
       78.45               89.47
       75.44               88.40
       66.57               58.38
       57.67    233.45     35.33
       23.91     87.88     31.04
       18.43     48.30     26.48
     +  7.32   + 11.55   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

65 :

                           88.40
       78.45               72.94
       75.44               67.37
       66.57               58.38
       57.67    233.45     49.30
       23.91     87.88     42.81
       18.43     48.30     34.60
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

66 :

                           91.36
       78.45               88.40
       75.44               72.94
       66.57               49.30
       57.67    233.45     42.81
       23.91     87.88     35.33
       18.43     48.30     31.04
     +  7.32   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

67 :

       78.45     91.36
       75.44     88.40    234.65
       66.57     72.94     98.43
       57.67     42.81     48.30
       23.91     35.33     34.60
       18.43     26.48     11.55
     +  7.32   + 23.86   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

68 :

                           89.47
       78.45               87.88
       75.44               72.94
       66.57               67.37
       57.67               42.81
       23.91    233.45     34.60
       18.43     98.43     31.04
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

69 :

       78.45               91.36
       75.44               88.40
       66.57               72.94
       57.67               67.37
       23.91    233.45     48.30
       18.43     98.43     42.81
     +  7.32   + 49.30   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

70 :

                           88.40
       91.36               87.88
       78.45     98.43     67.37
       58.38     75.44     49.30
       42.81     72.94     48.30
       31.04     66.57     35.33
       18.43     57.67     34.60
     +  7.32   + 10.13   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

71 :

       91.36               98.43
       78.45               95.02
       58.38               89.47
       42.81    233.45     57.67
       31.04     87.88     49.30
       18.43     48.30     23.91
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

72 :

       91.36               89.47
       78.45               88.40
       58.38               72.94
       42.81    233.45     67.37
       31.04     87.88     57.67
       18.43     48.30     35.33
     +  7.32   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

73 :

       91.36               95.02
       78.45               89.47
       58.38               67.37
       42.81    233.45     66.57
       31.04     87.88     49.30
       18.43     48.30     35.33
     +  7.32   + 11.55   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

74 :

                           95.02
                           88.40
       91.36               75.44
       78.45               57.67
       58.38               35.33
       42.81               26.48
       31.04    233.45     23.91
       18.43     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

75 :

                           95.02
       91.36               89.47
       78.45               87.88
       58.38               57.67
       42.81               48.30
       31.04    233.45     23.91
       18.43     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

76 :

                           89.47
       91.36               88.40
       78.45               87.88
       58.38               66.57
       42.81               35.33
       31.04    233.45     34.60
       18.43     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

77 :

                           89.47
       91.36               75.44
       78.45               72.94
       58.38               66.57
       42.81               48.30
       31.04    233.45     34.60
       18.43     98.43     26.48
     +  7.32   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

78 :

                           89.47
       91.36               75.44
       78.45               72.94
       58.38               57.67
       42.81               48.30
       31.04    233.45     35.33
       18.43     98.43     34.60
     +  7.32   + 49.30   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

79 :

       91.36               89.47
       78.45               88.40
       58.38               72.94
       42.81               67.37
       31.04    233.45     57.67
       18.43     98.43     35.33
     +  7.32   + 49.30   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

80 :

       87.88     98.43     91.36
       72.94     78.45     88.40
       67.37     66.57     75.44
       42.81     57.67     58.38
       31.04     35.33     49.30
       18.43     34.60     48.30
     +  7.32   + 10.13   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

81 :

       87.88     95.02
       72.94     91.36
       67.37     75.44    233.45
       42.81     49.30     78.45
       31.04     34.60     66.57
       18.43     23.91     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

82 :

       87.88     91.36
       72.94     88.40
       67.37     57.67    233.45
       42.81     49.30     78.45
       31.04     48.30     66.57
       18.43     34.60     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

83 :

       87.88
       72.94
       67.37     98.43    233.45
       42.81     91.36     78.45
       31.04     89.47     66.57
       18.43     75.44     35.33
     +  7.32   + 26.48   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

84 :

                           95.02
                           88.40
       87.88               75.44
       72.94               57.67
       67.37               35.33
       42.81               26.48
       31.04    233.45     23.91
       18.43     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

85 :

                           89.47
                           78.45
       87.88               66.57
       72.94               58.38
       67.37               48.30
       42.81               34.60
       31.04    233.45     26.48
       18.43     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

86 :

                           89.47
                           78.45
       87.88               58.38
       72.94               57.67
       67.37               48.30
       42.81               35.33
       31.04    233.45     34.60
       18.43     98.43     23.91
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

87 :

                           89.47
       87.88               78.45
       72.94               75.44
       67.37               66.57
       42.81               57.67
       31.04    233.45     34.60
       18.43     98.43     23.91
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

88 :

       87.88     98.43     91.36
       75.44     78.45     88.40
       58.38     66.57     72.94
       49.30     57.67     67.37
       31.04     35.33     48.30
       18.43     34.60     42.81
     +  7.32   + 10.13   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

89 :

       87.88     89.47
       75.44     88.40
       58.38     67.37    233.45
       49.30     57.67     78.45
       31.04     42.81     66.57
       18.43     23.91     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

90 :

       87.88     91.36
       75.44     88.40    234.65
       58.38     72.94     98.43
       49.30     42.81     48.30
       31.04     35.33     34.60
       18.43     26.48     11.55
     +  7.32   + 23.86   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

91 :

       78.45     98.43     88.40
       72.94     91.36     87.88
       58.38     75.44     67.37
       57.67     48.30     66.57
       34.60     31.04     49.30
       18.43     26.48     42.81
     +  7.32   + 10.13   + 35.33
     =======   =======   =======
      327.79    381.18    437.66

92 :

                           88.40
       78.45               75.44
       72.94               67.37
       58.38               66.57
       57.67    233.45     49.30
       34.60     87.88     42.81
       18.43     48.30     23.91
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

93 :

                           98.43
       78.45               89.47
       72.94               67.37
       58.38               49.30
       57.67    233.45     42.81
       34.60     87.88     35.33
       18.43     48.30     31.04
     +  7.32   + 11.55   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

94 :

       78.45               88.40
       72.94               87.88
       58.38     98.43     67.37
       57.67     91.36     66.57
       34.60     89.47     49.30
       18.43     75.44     42.81
     +  7.32   + 26.48   + 35.33
     =======   =======   =======
      327.79    381.18    437.66

95 :

                           89.47
                           87.88
       78.45               67.37
       72.94               48.30
       58.38               42.81
       57.67               35.33
       34.60    233.45     31.04
       18.43     98.43     23.91
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

96 :

       78.45
       72.94              234.65
       58.38               67.37
       57.67               66.57
       34.60    233.45     31.04
       18.43     98.43     26.48
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

97 :

                           91.36
       78.45               89.47
       72.94               75.44
       58.38               48.30
       57.67               42.81
       34.60    233.45     35.33
       18.43     98.43     31.04
     +  7.32   + 49.30   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

98 :

       89.47
       48.30
       42.81
       34.60     98.43    234.65
       31.04     75.44     88.40
       26.48     72.94     49.30
       23.91     66.57     35.33
       23.86     57.67     18.43
     +  7.32   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

99 :

       89.47
       48.30
       42.81               95.02
       34.60               88.40
       31.04               75.44
       26.48               67.37
       23.91    233.45     57.67
       23.86     98.43     35.33
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

100 :

       95.02
       66.57               89.47
       49.30               88.40
       35.33               75.44
       26.48    233.45     72.94
       23.91     87.88     58.38
       23.86     48.30     34.60
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

101 :

       95.02
       66.57               91.36
       49.30               89.47
       35.33               75.44
       26.48    233.45     72.94
       23.91     87.88     42.81
       23.86     48.30     34.60
     +  7.32   + 11.55   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

102 :

                           88.40
       87.88               78.45
       58.38               67.37
       48.30               66.57
       42.81               57.67
       35.33               31.04
       23.91    233.45     26.48
       23.86     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

103 :

       87.88
       58.38
       48.30              234.65
       42.81               67.37
       35.33               66.57
       23.91    233.45     31.04
       23.86     98.43     26.48
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

104 :

       87.88
       75.44     98.43    234.65
       66.57     91.36     88.40
       42.81     67.37     49.30
       23.91     58.38     35.33
       23.86     34.60     18.43
     +  7.32   + 31.04   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

105 :

                           91.36
                           89.47
       87.88               67.37
       75.44               57.67
       66.57               48.30
       42.81               35.33
       23.91    233.45     26.48
       23.86     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

106 :

                           88.40
                           72.94
       87.88               67.37
       75.44               57.67
       66.57               48.30
       42.81               35.33
       23.91    233.45     31.04
       23.86     98.43     26.48
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

107 :

                           91.36
       87.88               89.47
       75.44               72.94
       66.57               57.67
       42.81               48.30
       23.91    233.45     35.33
       23.86     98.43     31.04
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

108 :

       87.88               89.47
       75.44               88.40
       66.57               72.94
       42.81               67.37
       23.91    233.45     57.67
       23.86     98.43     35.33
     +  7.32   + 49.30   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

109 :

       98.43     89.47     95.02
       91.36     87.88     88.40
       49.30     66.57     75.44
       31.04     48.30     67.37
       26.48     42.81     57.67
       23.86     34.60     35.33
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

110 :

       98.43               89.47
       91.36               88.40
       49.30               75.44
       31.04    233.45     72.94
       26.48     87.88     58.38
       23.86     48.30     34.60
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

111 :

       98.43               95.02
       91.36               88.40
       49.30               75.44
       31.04    233.45     67.37
       26.48     87.88     57.67
       23.86     48.30     35.33
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

112 :

       87.88     98.43
       72.94     91.36    234.65
       58.38     75.44     88.40
       42.81     48.30     49.30
       34.60     31.04     35.33
       23.86     26.48     18.43
     +  7.32   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

113 :

       87.88
       72.94              234.65
       58.38     98.43     88.40
       42.81     91.36     49.30
       34.60     89.47     35.33
       23.86     75.44     18.43
     +  7.32   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

114 :

                           88.40
                           78.45
       87.88               67.37
       72.94               66.57
       58.38               57.67
       42.81               31.04
       34.60    233.45     26.48
       23.86     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

115 :

                           91.36
                           89.47
       87.88               67.37
       72.94               57.67
       58.38               48.30
       42.81               35.33
       34.60    233.45     26.48
       23.86     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

116 :

                           89.47
                           78.45
       87.88               67.37
       72.94               66.57
       58.38               48.30
       42.81               31.04
       34.60    233.45     26.48
       23.86     98.43     18.43
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

117 :

       87.88
       72.94              234.65
       58.38               67.37
       42.81               66.57
       34.60    233.45     31.04
       23.86     98.43     26.48
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

118 :

       87.88               95.02
       72.94               88.40
       58.38               75.44
       42.81               67.37
       34.60    233.45     57.67
       23.86     98.43     35.33
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

119 :

                 98.43
       89.47     88.40
       66.57     72.94    234.65
       57.67     35.33     75.44
       48.30     31.04     49.30
       34.60     26.48     42.81
       23.86     18.43     23.91
     +  7.32   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

120 :

                 98.43
       89.47     75.44
       66.57     72.94    234.65
       57.67     42.81     88.40
       48.30     31.04     49.30
       34.60     26.48     35.33
       23.86     23.91     18.43
     +  7.32   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

121 :

       89.47
       66.57     98.43    234.65
       57.67     88.40     75.44
       48.30     78.45     49.30
       34.60     58.38     42.81
       23.86     31.04     23.91
     +  7.32   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

122 :

                           95.02
                           78.45
                           67.37
       89.47               58.38
       66.57               35.33
       57.67               31.04
       48.30               26.48
       34.60    233.45     23.91
       23.86     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

123 :

                           88.40
                           87.88
       89.47               78.45
       66.57               72.94
       57.67               31.04
       48.30               26.48
       34.60    233.45     23.91
       23.86     98.43     18.43
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

124 :

                           95.02
                           75.44
       89.47               72.94
       66.57               67.37
       57.67               35.33
       48.30               31.04
       34.60    233.45     26.48
       23.86     98.43     23.91
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

125 :

       66.57
       58.38
       57.67              234.65
       48.30     98.43     88.40
       34.60     91.36     49.30
       31.04     89.47     35.33
       23.91     75.44     18.43
     +  7.32   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

126 :

                           95.02
       66.57               89.47
       58.38               78.45
       57.67               67.37
       48.30               35.33
       34.60               26.48
       31.04    233.45     23.86
       23.91     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

127 :

                 98.43
       88.40     87.88
       58.38     67.37
       57.67     48.30    233.45
       49.30     31.04     78.45
       42.81     26.48     66.57
       23.91     11.55     35.33
     +  7.32   + 10.13   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

128 :

       88.40     98.43
       58.38     91.36
       57.67     75.44    233.45
       49.30     48.30     78.45
       42.81     31.04     66.57
       23.91     26.48     35.33
     +  7.32   + 10.13   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

129 :

       88.40     89.47
       58.38     87.88
       57.67     75.44    233.45
       49.30     67.37     78.45
       42.81     31.04     66.57
       23.91     18.43     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

130 :

       88.40
       58.38     98.43
       57.67     89.47    233.45
       49.30     87.88     78.45
       42.81     67.37     66.57
       23.91     26.48     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

131 :

                           98.43
                           78.45
       88.40               67.37
       58.38               66.57
       57.67               35.33
       49.30    233.45     31.04
       42.81     87.88     26.48
       23.91     48.30     23.86
     +  7.32   + 11.55   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

132 :

                           78.45
       88.40               75.44
       58.38               72.94
       57.67               67.37
       49.30    233.45     66.57
       42.81     87.88     34.60
       23.91     48.30     23.86
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

133 :

       88.40
       58.38
       57.67     98.43    233.45
       49.30     91.36     78.45
       42.81     89.47     66.57
       23.91     75.44     35.33
     +  7.32   + 26.48   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

134 :

       95.02     98.43    234.65
       91.36     75.44     88.40
       67.37     72.94     49.30
       42.81     66.57     35.33
       23.91     57.67     18.43
     +  7.32   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

135 :

                 87.88
                 75.44
       95.02     72.94
       91.36     49.30    233.45
       67.37     34.60     78.45
       42.81     31.04     66.57
       23.91     18.43     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

136 :

                 98.43
       95.02     87.88
       91.36     72.94    233.45
       67.37     49.30     78.45
       42.81     34.60     66.57
       23.91     26.48     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

137 :

                           98.43
                           88.40
                           66.57
       95.02               49.30
       91.36               35.33
       67.37    233.45     34.60
       42.81     87.88     31.04
       23.91     48.30     23.86
     +  7.32   + 11.55   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

138 :

                           98.43
                           89.47
       95.02               88.40
       91.36               58.38
       67.37    233.45     35.33
       42.81     87.88     31.04
       23.91     48.30     26.48
     +  7.32   + 11.55   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

139 :

                           89.47
       95.02               88.40
       91.36               75.44
       67.37    233.45     72.94
       42.81     87.88     58.38
       23.91     48.30     34.60
     +  7.32   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

140 :

                           89.47
                           72.94
                           66.57
                           48.30
                           35.33
       95.02               34.60
       91.36               26.48
       67.37               23.86
       42.81    233.45     18.43
       23.91     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

141 :

                           88.40
                           87.88
                           66.57
                           48.30
       95.02               35.33
       91.36               34.60
       67.37               31.04
       42.81    233.45     23.86
       23.91     98.43     11.55
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

142 :

                           89.47
                           87.88
                           75.44
       95.02               57.67
       91.36               48.30
       67.37               26.48
       42.81    233.45     23.86
       23.91     98.43     18.43
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

143 :

                           88.40
                           75.44
                           72.94
       95.02               58.38
       91.36               48.30
       67.37               34.60
       42.81    233.45     31.04
       23.91     98.43     18.43
     +  7.32   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

144 :

                           87.88
                           78.45
                           72.94
       95.02               66.57
       91.36               35.33
       67.37               34.60
       42.81    233.45     26.48
       23.91     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

145 :

                           89.47
                           78.45
                           66.57
       95.02               58.38
       91.36               48.30
       67.37               34.60
       42.81    233.45     26.48
       23.91     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

146 :

                           89.47
                           88.40
       95.02               87.88
       91.36               66.57
       67.37               35.33
       42.81    233.45     34.60
       23.91     98.43     23.86
     +  7.32   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

147 :

                           89.47
       95.02               88.40
       91.36               75.44
       67.37               72.94
       42.81    233.45     58.38
       23.91     98.43     34.60
     +  7.32   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

148 :

                           89.47
                           75.44
       95.02               72.94
       91.36               66.57
       67.37               48.30
       42.81    233.45     34.60
       23.91     98.43     26.48
     +  7.32   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

149 :

                           88.40
       95.02               87.88
       91.36               78.45
       67.37               75.44
       42.81    233.45     48.30
       23.91     98.43     35.33
     +  7.32   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

150 :

                 88.40
                 67.37
                 57.67
       98.43     48.30
       87.88     42.81    233.45
       58.38     31.04     78.45
       49.30     23.91     66.57
       26.48     11.55     35.33
     +  7.32   + 10.13   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

151 :

                 89.47
       98.43     88.40
       87.88     67.37    233.45
       58.38     57.67     78.45
       49.30     42.81     66.57
       26.48     23.91     35.33
     +  7.32   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

152 :

       98.43     95.02
       87.88     88.40    233.45
       58.38     75.44     78.45
       49.30     67.37     66.57
       26.48     31.04     35.33
     +  7.32   + 23.91   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

153 :

       67.37
       66.57
       57.67               91.36
       48.30               89.47
       23.91               75.44
       23.86               72.94
       18.43    233.45     42.81
       11.55     98.43     34.60
     + 10.13   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

154 :

       87.88
       66.57
       42.81               95.02
       34.60               88.40
       26.48               75.44
       23.91               67.37
       23.86    233.45     57.67
       11.55     98.43     35.33
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

155 :

       95.02               91.36
       89.47               87.88
       42.81               72.94
       31.04               57.67
       23.91               48.30
       23.86    233.45     34.60
       11.55     98.43     26.48
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

156 :

       95.02               89.47
       87.88               88.40
       75.44    233.45     72.94
       23.91     49.30     67.37
       23.86     48.30     57.67
       11.55     42.81     35.33
     + 10.13   +  7.32   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

157 :

                           88.40
                           67.37
       95.02               66.57
       87.88               57.67
       75.44               42.81
       23.91               35.33
       23.86    233.45     34.60
       11.55     98.43     26.48
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

158 :

       95.02               89.47
       87.88               88.40
       75.44               72.94
       23.91               67.37
       23.86    233.45     57.67
       11.55     98.43     35.33
     + 10.13   + 49.30   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

159 :

       95.02               91.36
       87.88               88.40
       75.44               72.94
       23.91               67.37
       23.86    233.45     48.30
       11.55     98.43     42.81
     + 10.13   + 49.30   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

160 :

                           98.43
       88.40               89.47
       87.88     95.02     67.37
       57.67     78.45     49.30
       48.30     75.44     42.81
       23.86     66.57     35.33
       11.55     58.38     31.04
     + 10.13   +  7.32   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

161 :

                           95.02
                           78.45
       88.40               75.44
       87.88               58.38
       57.67               42.81
       48.30               35.33
       23.86    233.45     26.48
       11.55     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

162 :

                           78.45
                           67.37
                           66.57
       88.40               58.38
       87.88               42.81
       57.67               35.33
       48.30               31.04
       23.86    233.45     26.48
       11.55     98.43     23.91
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

163 :

                           95.02
                           75.44
       88.40               66.57
       87.88               58.38
       57.67               42.81
       48.30               34.60
       23.86    233.45     31.04
       11.55     98.43     26.48
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

164 :

       88.40               91.36
       87.88               89.47
       57.67               75.44
       48.30               72.94
       23.86    233.45     42.81
       11.55     98.43     34.60
     + 10.13   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

165 :

       78.45
       75.44               95.02
       58.38               91.36
       35.33               87.88
       34.60               67.37
       23.91    233.45     57.67
       11.55     98.43     31.04
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

166 :

       78.45
       75.44               91.36
       58.38               87.88
       35.33               72.94
       34.60               66.57
       23.91    233.45     57.67
       11.55     98.43     42.81
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

167 :

       78.45
       75.44               91.36
       58.38               88.40
       35.33               72.94
       34.60               67.37
       23.91    233.45     48.30
       11.55     98.43     42.81
     + 10.13   + 49.30   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

168 :

       87.88
       78.45              234.65
       66.57     98.43     95.02
       49.30     91.36     58.38
       23.91     89.47     23.86
       11.55     75.44     18.43
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

169 :

       89.47
       67.37
       48.30              234.65
       35.33               95.02
       34.60               58.38
       31.04    233.45     23.86
       11.55     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

170 :

                           95.02
       89.47               78.45
       67.37               66.57
       48.30               58.38
       35.33               57.67
       34.60               26.48
       31.04    233.45     23.91
       11.55     98.43     23.86
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

171 :

       89.47
       67.37               91.36
       48.30               87.88
       35.33               72.94
       34.60               66.57
       31.04    233.45     57.67
       11.55     98.43     42.81
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

172 :

                           89.47
       91.36               87.88
       75.44               67.37
       72.94    233.45     58.38
       35.33     49.30     57.67
       31.04     48.30     34.60
       11.55     42.81     23.86
     + 10.13   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

173 :

       91.36               95.02
       75.44               89.47
       72.94    233.45     88.40
       35.33     49.30     87.88
       31.04     48.30     34.60
       11.55     42.81     23.86
     + 10.13   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

174 :

       91.36               98.43
       75.44               88.40
       72.94    233.45     67.37
       35.33     49.30     66.57
       31.04     48.30     58.38
       11.55     42.81     34.60
     + 10.13   +  7.32   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

175 :

       91.36
       75.44     98.43    234.65
       72.94     89.47     95.02
       35.33     66.57     58.38
       31.04     49.30     23.86
       11.55     42.81     18.43
     + 10.13   + 34.60   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

176 :

       91.36
       75.44              234.65
       72.94               95.02
       35.33               58.38
       31.04    233.45     23.86
       11.55     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

177 :

                           95.02
                           78.45
       91.36               66.57
       75.44               58.38
       72.94               57.67
       35.33               26.48
       31.04    233.45     23.91
       11.55     98.43     23.86
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

178 :

                           88.40
                           67.37
       91.36               66.57
       75.44               58.38
       72.94               48.30
       35.33               42.81
       31.04    233.45     34.60
       11.55     98.43     23.91
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

179 :

                           89.47
       91.36               87.88
       75.44               67.37
       72.94               58.38
       35.33               57.67
       31.04    233.45     34.60
       11.55     98.43     23.86
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

180 :

       91.36               95.02
       75.44               89.47
       72.94               88.40
       35.33               87.88
       31.04    233.45     34.60
       11.55     98.43     23.86
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

181 :

                           88.40
       95.02               75.44
       78.45               72.94
       66.57               58.38
       35.33    233.45     49.30
       23.86     87.88     42.81
       18.43     48.30     26.48
     + 10.13   + 11.55   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

182 :

       95.02               91.36
       78.45               89.47
       66.57               75.44
       35.33    233.45     72.94
       23.86     87.88     42.81
       18.43     48.30     34.60
     + 10.13   + 11.55   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

183 :

       95.02
       78.45              234.65
       66.57     98.43     87.88
       35.33     91.36     49.30
       23.86     89.47     34.60
       18.43     75.44     23.91
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

184 :

       95.02
       78.45     98.43    234.65
       66.57     91.36     75.44
       35.33     67.37     49.30
       23.86     58.38     42.81
       18.43     34.60     23.91
     + 10.13   + 31.04   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

185 :

                           91.36
       95.02               89.47
       78.45               87.88
       66.57               58.38
       35.33               48.30
       23.86    233.45     31.04
       18.43     98.43     23.91
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

186 :

                           89.47
       95.02               75.44
       78.45               72.94
       66.57               58.38
       35.33               57.67
       23.86    233.45     48.30
       18.43     98.43     23.91
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

187 :

                           89.47
       95.02               87.88
       78.45               72.94
       66.57               67.37
       35.33               42.81
       23.86    233.45     34.60
       18.43     98.43     31.04
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

188 :

       95.02               91.36
       78.45               88.40
       66.57               72.94
       35.33               67.37
       23.86    233.45     48.30
       18.43     98.43     42.81
     + 10.13   + 49.30   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

189 :

       95.02               91.36
       78.45               89.47
       66.57               75.44
       35.33               72.94
       23.86    233.45     42.81
       18.43     98.43     34.60
     + 10.13   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

190 :

       72.94
       66.57               91.36
       57.67               87.88
       42.81               78.45
       35.33               75.44
       23.91    233.45     58.38
       18.43     98.43     34.60
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

191 :

       91.36
       72.94               98.43
       42.81               95.02
       34.60               89.47
       31.04    233.45     57.67
       26.48     87.88     49.30
       18.43     48.30     23.91
     + 10.13   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

192 :

       91.36
       72.94              234.65
       42.81               66.57
       34.60               58.38
       31.04               35.33
       26.48    233.45     23.86
       18.43     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

193 :

       91.36               95.02
       72.94               89.47
       42.81               87.88
       34.60               57.67
       31.04               48.30
       26.48    233.45     23.91
       18.43     98.43     23.86
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

194 :

       91.36
       72.94               88.40
       42.81               87.88
       34.60               78.45
       31.04               75.44
       26.48    233.45     48.30
       18.43     98.43     35.33
     + 10.13   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

195 :

       87.88     98.43     91.36
       67.37     78.45     88.40
       66.57     72.94     75.44
       42.81     57.67     58.38
       34.60     35.33     49.30
       18.43     31.04     48.30
     + 10.13   +  7.32   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

196 :

                           91.36
       87.88               75.44
       67.37     98.43     72.94
       66.57     88.40     57.67
       42.81     78.45     49.30
       34.60     58.38     48.30
       18.43     31.04     35.33
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

197 :

       87.88
       67.37              233.45
       66.57     98.43     88.40
       42.81     91.36     49.30
       34.60     89.47     35.33
       18.43     75.44     23.86
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

198 :

                           91.36
       87.88               89.47
       67.37               88.40
       66.57               75.44
       42.81               35.33
       34.60    233.45     26.48
       18.43     98.43     23.86
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

199 :

                           91.36
                           88.40
       87.88               75.44
       67.37               58.38
       66.57               35.33
       42.81               31.04
       34.60    233.45     26.48
       18.43     98.43     23.91
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

200 :

                           95.02
                           88.40
       87.88               75.44
       67.37               57.67
       66.57               35.33
       42.81               26.48
       34.60    233.45     23.91
       18.43     98.43     23.86
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

201 :

                           95.02
       87.88               89.47
       67.37               78.45
       66.57               72.94
       42.81               35.33
       34.60    233.45     31.04
       18.43     98.43     23.86
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

202 :

                           89.47
       87.88               75.44
       67.37               72.94
       66.57               58.38
       42.81               57.67
       34.60    233.45     48.30
       18.43     98.43     23.91
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

203 :

                           91.36
       87.88               89.47
       67.37               72.94
       66.57               57.67
       42.81               48.30
       34.60    233.45     35.33
       18.43     98.43     31.04
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

204 :

       87.88               91.36
       67.37               89.47
       66.57               78.45
       42.81               72.94
       34.60    233.45     57.67
       18.43     98.43     23.91
     + 10.13   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

205 :

       91.36
       66.57     98.43    234.65
       57.67     88.40     87.88
       48.30     78.45     49.30
       35.33     58.38     34.60
       18.43     31.04     23.91
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

206 :

                           87.88
       91.36               75.44
       66.57     98.43     72.94
       57.67     88.40     67.37
       48.30     78.45     49.30
       35.33     58.38     42.81
       18.43     31.04     34.60
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

207 :

       91.36
       66.57     98.43    234.65
       57.67     88.40     75.44
       48.30     78.45     49.30
       35.33     58.38     42.81
       18.43     31.04     23.91
     + 10.13   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

208 :

                           98.43
       91.36               88.40
       66.57     87.88     78.45
       57.67     75.44     72.94
       48.30     67.37     34.60
       35.33     58.38     31.04
       18.43     49.30     26.48
     + 10.13   + 42.81   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

209 :

                           95.02
                           88.40
       91.36               78.45
       66.57               67.37
       57.67               34.60
       48.30               31.04
       35.33    233.45     23.91
       18.43     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

210 :

                           89.47
       91.36               87.88
       66.57               72.94
       57.67               67.37
       48.30               42.81
       35.33    233.45     34.60
       18.43     98.43     31.04
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

211 :

                           98.43
       89.47               75.44
       78.45               66.57
       67.37               58.38
       34.60    233.45     57.67
       23.91     87.88     42.81
       23.86     48.30     31.04
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

212 :

                           91.36
       89.47               88.40
       78.45               72.94
       67.37               49.30
       34.60    233.45     42.81
       23.91     87.88     35.33
       23.86     48.30     31.04
     + 10.13   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

213 :

                           95.02
                           66.57
                           58.38
       89.47               57.67
       78.45               48.30
       67.37               35.33
       34.60               31.04
       23.91    233.45     26.48
       23.86     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

214 :

       89.47               91.36
       78.45               87.88
       67.37               72.94
       34.60               66.57
       23.91    233.45     57.67
       23.86     98.43     42.81
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

215 :

       67.37
       58.38               98.43
       57.67               91.36
       49.30               89.47
       34.60    233.45     72.94
       26.48     87.88     42.81
       23.86     48.30     35.33
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

216 :

                           98.43
       95.02               75.44
       88.40               66.57
       49.30               58.38
       34.60    233.45     57.67
       26.48     87.88     42.81
       23.86     48.30     31.04
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

217 :

       95.02               98.43
       88.40               91.36
       49.30               89.47
       34.60    233.45     72.94
       26.48     87.88     42.81
       23.86     48.30     35.33
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

218 :

       95.02               98.43
       88.40               91.36
       49.30               78.45
       34.60    233.45     67.37
       26.48     87.88     42.81
       23.86     48.30     35.33
     + 10.13   + 11.55   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

219 :

       78.45     95.02     98.43
       66.57     89.47     87.88
       57.67     88.40     75.44
       48.30     58.38     67.37
       42.81     31.04     49.30
       23.86     11.55     35.33
     + 10.13   +  7.32   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

220 :

                           98.43
       78.45               89.47
       66.57     95.02     87.88
       57.67     88.40     58.38
       48.30     75.44     49.30
       42.81     67.37     35.33
       23.86     31.04     11.55
     + 10.13   + 23.91   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

221 :

       78.45
       66.57              234.65
       57.67     98.43     87.88
       48.30     91.36     49.30
       42.81     89.47     34.60
       23.86     75.44     23.91
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

222 :

       78.45
       66.57              234.65
       57.67     98.43     88.40
       48.30     91.36     49.30
       42.81     89.47     35.33
       23.86     75.44     18.43
     + 10.13   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

223 :

                           95.02
       78.45               88.40
       66.57               87.88
       57.67     98.43     58.38
       48.30     91.36     34.60
       42.81     89.47     31.04
       23.86     75.44     23.91
     + 10.13   + 26.48   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

224 :

       78.45
       66.57     98.43    234.65
       57.67     91.36     88.40
       48.30     67.37     49.30
       42.81     58.38     35.33
       23.86     34.60     18.43
     + 10.13   + 31.04   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

225 :

                           88.40
                           87.88
                           67.37
       78.45               58.38
       66.57               35.33
       57.67               31.04
       48.30               26.48
       42.81    233.45     23.91
       23.86     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

226 :

                           91.36
                           88.40
       78.45               75.44
       66.57               58.38
       57.67               35.33
       48.30               31.04
       42.81    233.45     26.48
       23.86     98.43     23.91
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

227 :

                           89.47
                           88.40
       78.45               75.44
       66.57               72.94
       57.67               31.04
       48.30               26.48
       42.81    233.45     23.91
       23.86     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

228 :

                           95.02
       78.45               88.40
       66.57               87.88
       57.67               58.38
       48.30               34.60
       42.81    233.45     31.04
       23.86     98.43     23.91
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

229 :

       78.45               89.47
       66.57               88.40
       57.67               75.44
       48.30               72.94
       42.81    233.45     58.38
       23.86     98.43     34.60
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

230 :

                           89.47
                           87.88
                           67.37
       98.43     95.02     48.30
       88.40     78.45     42.81
       57.67     75.44     35.33
       49.30     66.57     31.04
       23.86     58.38     23.91
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

231 :

                           89.47
                           87.88
       98.43     95.02     72.94
       88.40     78.45     67.37
       57.67     75.44     42.81
       49.30     66.57     34.60
       23.86     58.38     31.04
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

232 :

                           95.02
                           78.45
                           75.44
       98.43               58.38
       88.40               42.81
       57.67    233.45     35.33
       49.30     87.88     26.48
       23.86     48.30     18.43
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

233 :

                           78.45
                           67.37
                           66.57
                           58.38
       98.43               42.81
       88.40               35.33
       57.67    233.45     31.04
       49.30     87.88     26.48
       23.86     48.30     23.91
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

234 :

                           95.02
                           75.44
                           66.57
       98.43               58.38
       88.40               42.81
       57.67    233.45     34.60
       49.30     87.88     31.04
       23.86     48.30     26.48
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

235 :

                           91.36
       98.43               89.47
       88.40               75.44
       57.67    233.45     72.94
       49.30     87.88     42.81
       23.86     48.30     34.60
     + 10.13   + 11.55   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

236 :

       87.88
       72.94     98.43    234.65
       49.30     88.40     67.37
       48.30     75.44     66.57
       35.33     57.67     31.04
       23.91     42.81     26.48
     + 10.13   + 18.43   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

237 :

       87.88
       72.94              234.65
       49.30     98.43     95.02
       48.30     91.36     58.38
       35.33     89.47     23.86
       23.91     75.44     18.43
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

238 :

                 98.43
                 89.47
                 58.38
       95.02     34.60
       67.37     31.04    233.45
       57.67     26.48     78.45
       49.30     23.91     66.57
       48.30     11.55     35.33
     + 10.13   +  7.32   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

239 :

                 91.36
                 87.88     89.47
       95.02     78.45     88.40
       67.37     42.81     75.44
       57.67     35.33     72.94
       49.30     26.48     58.38
       48.30     11.55     34.60
     + 10.13   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

240 :

                 89.47
                 88.40     91.36
       95.02     72.94     87.88
       67.37     42.81     78.45
       57.67     35.33     75.44
       49.30     26.48     58.38
       48.30     18.43     34.60
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

241 :

                          233.45
       95.02     98.43     87.88
       67.37     88.40     42.81
       57.67     78.45     23.91
       49.30     58.38     23.86
       48.30     31.04     18.43
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

242 :

                          234.65
       95.02               66.57
       67.37     98.43     58.38
       57.67     91.36     35.33
       49.30     89.47     23.86
       48.30     75.44     11.55
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

243 :

                          233.45
       95.02               87.88
       67.37     98.43     42.81
       57.67     91.36     23.91
       49.30     89.47     23.86
       48.30     75.44     18.43
     + 10.13   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

244 :

                           88.40
                           78.45
                           72.94
       95.02               66.57
       67.37     98.43     42.81
       57.67     91.36     34.60
       49.30     89.47     23.91
       48.30     75.44     18.43
     + 10.13   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

245 :

                           88.40
                           87.88
                           66.57
       95.02               58.38
       67.37     98.43     35.33
       57.67     91.36     34.60
       49.30     89.47     31.04
       48.30     75.44     23.91
     + 10.13   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

246 :

       95.02
       67.37     98.43    233.45
       57.67     91.36     78.45
       49.30     89.47     66.57
       48.30     75.44     35.33
     + 10.13   + 26.48   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

247 :

                 95.02     88.40
                 75.44     87.88
                 72.94     67.37
       91.36     57.67     49.30
       89.47     42.81     48.30
       78.45     18.43     35.33
       58.38     11.55     34.60
     + 10.13   +  7.32   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

248 :

                 87.88
                 72.94
                 49.30     95.02
                 48.30     88.40
       91.36     42.81     75.44
       89.47     34.60     67.37
       78.45     26.48     57.67
       58.38     11.55     35.33
     + 10.13   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

249 :

                 95.02
                 72.94
                 67.37    234.65
       91.36     48.30     75.44
       89.47     35.33     49.30
       78.45     31.04     42.81
       58.38     23.86     23.91
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

250 :

                 95.02
                 87.88
                 57.67    234.65
       91.36     48.30     88.40
       89.47     34.60     49.30
       78.45     26.48     35.33
       58.38     23.91     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

251 :

                 75.44
                 67.37
                 66.57    234.65
       91.36     57.67     88.40
       89.47     48.30     49.30
       78.45     34.60     35.33
       58.38     23.91     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

252 :

                 95.02
                 87.88    234.65
       91.36     66.57     88.40
       89.47     57.67     49.30
       78.45     42.81     35.33
       58.38     23.91     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

253 :

                           98.43
                           95.02
                           88.40
       91.36    233.45     67.37
       89.47     49.30     34.60
       78.45     48.30     23.86
       58.38     42.81     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

254 :

                           98.43
                           95.02
                           88.40
       91.36    233.45     75.44
       89.47     49.30     26.48
       78.45     48.30     23.91
       58.38     42.81     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

255 :

                           95.02
                           88.40
                           75.44
                           57.67
       91.36    233.45     35.33
       89.47     49.30     26.48
       78.45     48.30     23.91
       58.38     42.81     23.86
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

256 :

                          234.65
       91.36    233.45     67.37
       89.47     49.30     66.57
       78.45     48.30     31.04
       58.38     42.81     26.48
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

257 :

                           98.43
                           75.44
                           67.37
                           57.67
       91.36    233.45     35.33
       89.47     49.30     34.60
       78.45     48.30     26.48
       58.38     42.81     23.91
     + 10.13   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

258 :

                           95.02
                           88.40
       91.36    233.45     75.44
       89.47     49.30     67.37
       78.45     48.30     57.67
       58.38     42.81     35.33
     + 10.13   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

259 :

                 95.02
                 88.40
                 57.67    234.65
       91.36     48.30     87.88
       89.47     35.33     49.30
       78.45     26.48     34.60
       58.38     18.43     23.91
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

260 :

                 95.02     87.88
                 88.40     75.44
                 57.67     72.94
       91.36     48.30     67.37
       89.47     35.33     49.30
       78.45     26.48     42.81
       58.38     18.43     34.60
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

261 :

                 95.02    233.45
                 72.94     57.67
       91.36     67.37     42.81
       89.47     66.57     35.33
       78.45     49.30     34.60
       58.38     18.43     26.48
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

262 :

                 95.02
                 75.44
                 57.67    233.45
       91.36     48.30     88.40
       89.47     42.81     49.30
       78.45     26.48     35.33
       58.38     23.91     23.86
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

263 :

                 88.40
                 75.44    233.45
                 66.57     57.67
       91.36     42.81     49.30
       89.47     35.33     31.04
       78.45     34.60     23.91
       58.38     26.48     23.86
     + 10.13   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

264 :

                 98.43     95.02
                 87.88     88.40
       91.36     72.94     75.44
       89.47     49.30     67.37
       78.45     34.60     57.67
       58.38     26.48     35.33
     + 10.13   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

265 :

                           88.40
                           75.44
                           72.94
                           67.37
       91.36               49.30
       89.47    233.45     34.60
       78.45     87.88     23.86
       58.38     48.30     18.43
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

266 :

                           95.02
                           88.40
                           66.57
                           57.67
       91.36               49.30
       89.47    233.45     31.04
       78.45     87.88     23.91
       58.38     48.30     18.43
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

267 :

                           98.43
                           75.44
                           67.37
                           57.67
       91.36               35.33
       89.47    233.45     34.60
       78.45     87.88     26.48
       58.38     48.30     23.91
     + 10.13   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

268 :

                           88.40
                           67.37
                           66.57
                           57.67
       91.36               42.81
       89.47    233.45     35.33
       78.45     87.88     34.60
       58.38     48.30     26.48
     + 10.13   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

269 :

                           95.02
                           88.40
       91.36               75.44
       89.47    233.45     67.37
       78.45     87.88     57.67
       58.38     48.30     35.33
     + 10.13   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

270 :

                           88.40
                           75.44
                           67.37
       91.36               66.57
       89.47    233.45     49.30
       78.45     87.88     42.81
       58.38     48.30     23.91
     + 10.13   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

271 :

                 98.43    233.45
                 75.44     57.67
       91.36     67.37     42.81
       89.47     49.30     35.33
       78.45     48.30     34.60
       58.38     23.91     26.48
     + 10.13   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

272 :

                           87.88
                           72.94
                 95.02     67.37
                 88.40     57.67
       91.36     75.44     49.30
       89.47     42.81     48.30
       78.45     34.60     35.33
       58.38     26.48     11.55
     + 10.13   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

273 :

                           98.43
                 95.02     87.88
                 88.40     66.57
       91.36     75.44     57.67
       89.47     42.81     48.30
       78.45     34.60     31.04
       58.38     26.48     23.91
     + 10.13   + 18.43   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

274 :

                 98.43    234.65
       91.36     88.40     87.88
       89.47     75.44     49.30
       78.45     57.67     34.60
       58.38     42.81     23.91
     + 10.13   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

275 :

                 98.43    234.65
       91.36     88.40     67.37
       89.47     75.44     66.57
       78.45     57.67     31.04
       58.38     42.81     26.48
     + 10.13   + 18.43   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

276 :

                           95.02
                           87.88
                           66.57
                 98.43     48.30
       91.36     88.40     34.60
       89.47     75.44     31.04
       78.45     57.67     26.48
       58.38     42.81     23.91
     + 10.13   + 18.43   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

277 :

                 98.43
                 72.94
                 49.30    234.65
       91.36     42.81     67.37
       89.47     35.33     66.57
       78.45     34.60     31.04
       58.38     23.91     26.48
     + 10.13   + 23.86   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

278 :

                          233.45
                 95.02     57.67
       91.36     88.40     42.81
       89.47     75.44     35.33
       78.45     67.37     34.60
       58.38     31.04     26.48
     + 10.13   + 23.91   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

279 :

                           95.02
                           88.40
                           67.37
                           48.30
                           42.81
       91.36               34.60
       89.47               23.86
       78.45    233.45     18.43
       58.38     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

280 :

                           95.02
                           88.40
                           75.44
                           48.30
                           42.81
       91.36               26.48
       89.47               23.91
       78.45    233.45     18.43
       58.38     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

281 :

                           95.02
                           87.88
                           66.57
                           48.30
                           35.33
       91.36               31.04
       89.47               23.91
       78.45    233.45     23.86
       58.38     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

282 :

                           95.02
                           87.88
                           72.94
                           66.57
       91.36               34.60
       89.47               31.04
       78.45    233.45     23.86
       58.38     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

283 :

                           75.44
                           67.37
                           57.67
                           48.30
                           42.81
                           35.33
       91.36               34.60
       89.47               26.48
       78.45    233.45     23.91
       58.38     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

284 :

                           95.02
                           75.44
                           72.94
                           48.30
                           34.60
       91.36               31.04
       89.47               26.48
       78.45    233.45     23.86
       58.38     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

285 :

                           95.02
                           75.44
                           72.94
                           66.57
       91.36               42.81
       89.47               31.04
       78.45    233.45     23.86
       58.38     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

286 :

                           95.02
                           88.40
                           75.44
                           57.67
       91.36               35.33
       89.47               26.48
       78.45    233.45     23.91
       58.38     98.43     23.86
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

287 :

                          234.65
       91.36               67.37
       89.47               66.57
       78.45    233.45     31.04
       58.38     98.43     26.48
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

288 :

                           88.40
                           67.37
                           66.57
                           57.67
       91.36               42.81
       89.47               35.33
       78.45    233.45     34.60
       58.38     98.43     26.48
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

289 :

                           95.02
                           88.40
       91.36               75.44
       89.47               67.37
       78.45    233.45     57.67
       58.38     98.43     35.33
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

290 :

                           95.02
                           87.88
                           66.57
                           48.30
       91.36               34.60
       89.47               31.04
       78.45    233.45     26.48
       58.38     98.43     23.91
     + 10.13   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

291 :

                 95.02
                 78.45
                 57.67     91.36
                 42.81     88.40
       89.47     35.33     75.44
       87.88     34.60     58.38
       72.94     18.43     49.30
       67.37     11.55     48.30
     + 10.13   +  7.32   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

292 :

                           91.36
                           75.44
                           58.38
                           57.67
                           35.33
       89.47    233.45     34.60
       87.88     49.30     31.04
       72.94     48.30     23.86
       67.37     42.81     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

293 :

                           98.43
                           95.02
                           88.40
       89.47    233.45     75.44
       87.88     49.30     26.48
       72.94     48.30     23.91
       67.37     42.81     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

294 :

                           95.02
                           88.40
                           75.44
                           57.67
       89.47    233.45     35.33
       87.88     49.30     26.48
       72.94     48.30     23.91
       67.37     42.81     23.86
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

295 :

                           98.43
                           91.36
                           58.38
                           57.67
       89.47    233.45     35.33
       87.88     49.30     34.60
       72.94     48.30     26.48
       67.37     42.81     23.86
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

296 :

                 95.02    234.65
       89.47     78.45     88.40
       87.88     75.44     49.30
       72.94     66.57     35.33
       67.37     58.38     18.43
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

297 :

                           98.43
                           88.40
                           57.67
                 95.02     49.30
       89.47     78.45     42.81
       87.88     75.44     34.60
       72.94     66.57     31.04
       67.37     58.38     23.86
     + 10.13   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

298 :

                          233.45
                 95.02     57.67
       89.47     78.45     49.30
       87.88     75.44     31.04
       72.94     66.57     23.91
       67.37     58.38     23.86
     + 10.13   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

299 :

                 95.02     91.36
                 88.40     78.45
                 57.67     75.44
       89.47     48.30     58.38
       87.88     35.33     49.30
       72.94     26.48     42.81
       67.37     18.43     34.60
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

300 :

                 98.43
                 95.02
                 49.30
       89.47     42.81    233.45
       87.88     34.60     78.45
       72.94     31.04     66.57
       67.37     18.43     35.33
     + 10.13   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

301 :

                           98.43
                 95.02     75.44
                 88.40     66.57
       89.47     78.45     58.38
       87.88     49.30     57.67
       72.94     34.60     42.81
       67.37     23.86     31.04
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

302 :

                 98.43    233.45
                 91.36     57.67
       89.47     58.38     42.81
       87.88     49.30     35.33
       72.94     48.30     34.60
       67.37     23.86     26.48
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

303 :

                 95.02
                 75.44
                 57.67    233.45
       89.47     48.30     88.40
       87.88     42.81     49.30
       72.94     26.48     35.33
       67.37     23.91     23.86
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

304 :

                 95.02
                 75.44
                 57.67
       89.47     48.30    233.45
       87.88     42.81     78.45
       72.94     26.48     66.57
       67.37     23.91     35.33
     + 10.13   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

305 :

                 95.02
                 91.36
       89.47     75.44    233.45
       87.88     49.30     78.45
       72.94     34.60     66.57
       67.37     23.91     35.33
     + 10.13   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

306 :

                 91.36
                 78.45
                 57.67    234.65
       89.47     49.30     95.02
       87.88     35.33     58.38
       72.94     31.04     23.86
       67.37     26.48     18.43
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

307 :

                 88.40
                 75.44
                 66.57    234.65
       89.47     42.81     95.02
       87.88     35.33     58.38
       72.94     34.60     23.86
       67.37     26.48     18.43
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

308 :

                 88.40
                 75.44    233.45
                 66.57     57.67
       89.47     42.81     49.30
       87.88     35.33     31.04
       72.94     34.60     23.91
       67.37     26.48     23.86
     + 10.13   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

309 :

                 91.36
                 88.40    234.65
       89.47     57.67     95.02
       87.88     49.30     58.38
       72.94     48.30     23.86
       67.37     34.60     18.43
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

310 :

                           95.02
                           78.45
                 91.36     75.44
                 88.40     58.38
       89.47     57.67     42.81
       87.88     49.30     35.33
       72.94     48.30     26.48
       67.37     34.60     18.43
     + 10.13   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

311 :

                 91.36
                 88.40
       89.47     57.67    233.45
       87.88     49.30     78.45
       72.94     48.30     66.57
       67.37     34.60     35.33
     + 10.13   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

312 :

                 95.02    234.65
                 88.40     66.57
       89.47     75.44     58.38
       87.88     42.81     35.33
       72.94     34.60     23.86
       67.37     26.48     11.55
     + 10.13   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

313 :

                           91.36
                           78.45
                 95.02     58.38
                 88.40     57.67
       89.47     75.44     49.30
       87.88     42.81     48.30
       72.94     34.60     35.33
       67.37     26.48     11.55
     + 10.13   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

314 :

                 95.02
                 88.40
       89.47     75.44    233.45
       87.88     42.81     78.45
       72.94     34.60     66.57
       67.37     26.48     35.33
     + 10.13   + 18.43   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

315 :

                          234.65
                 98.43     66.57
       89.47     88.40     58.38
       87.88     75.44     35.33
       72.94     57.67     23.86
       67.37     42.81     11.55
     + 10.13   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

316 :

                 98.43
       89.47     88.40    233.45
       87.88     75.44     78.45
       72.94     57.67     66.57
       67.37     42.81     35.33
     + 10.13   + 18.43   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

317 :

                 98.43    233.45
                 91.36     58.38
       89.47     49.30     57.67
       87.88     48.30     42.81
       72.94     35.33     26.48
       67.37     34.60     11.55
     + 10.13   + 23.86   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

318 :

                 78.45
                 75.44
                 57.67    234.65
       89.47     49.30     95.02
       87.88     35.33     58.38
       72.94     34.60     23.86
       67.37     26.48     18.43
     + 10.13   + 23.91   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

319 :

                 98.43    234.65
       89.47     88.40     75.44
       87.88     78.45     49.30
       72.94     58.38     42.81
       67.37     31.04     23.91
     + 10.13   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

320 :

                           95.02
                           88.40
                           75.44
                           48.30
                           42.81
       89.47               26.48
       87.88               23.91
       72.94    233.45     18.43
       67.37     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

321 :

                           91.36
                           58.38
                           57.67
                           48.30
                           42.81
                           35.33
       89.47               34.60
       87.88               26.48
       72.94    233.45     23.86
       67.37     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

322 :

                           91.36
                           75.44
                           66.57
                           57.67
                           42.81
       89.47               34.60
       87.88               26.48
       72.94    233.45     23.86
       67.37     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

323 :

                          234.65
                           66.57
       89.47               58.38
       87.88               35.33
       72.94    233.45     23.86
       67.37     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

324 :

                           95.02
                           66.57
                           58.38
                           57.67
                           48.30
       89.47               35.33
       87.88               31.04
       72.94    233.45     26.48
       67.37     98.43     11.55
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

325 :

                          234.65
       89.47               95.02
       87.88               58.38
       72.94    233.45     23.86
       67.37     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

326 :

                           95.02
                           78.45
                           75.44
                           58.38
       89.47               42.81
       87.88               35.33
       72.94    233.45     26.48
       67.37     98.43     18.43
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

327 :

                           95.02
                           78.45
                           66.57
                           58.38
       89.47               57.67
       87.88               26.48
       72.94    233.45     23.91
       67.37     98.43     23.86
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

328 :

                           91.36
                           88.40
                           75.44
                           58.38
       89.47               35.33
       87.88               31.04
       72.94    233.45     26.48
       67.37     98.43     23.91
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

329 :

                           95.02
                           75.44
                           66.57
                           58.38
       89.47               42.81
       87.88               34.60
       72.94    233.45     31.04
       67.37     98.43     26.48
     + 10.13   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

330 :

                           95.02
                           78.45
                           66.57
                           42.81
                           35.33
       89.47               34.60
       87.88               31.04
       72.94    233.45     23.86
       67.37     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

331 :

                           91.36
                           75.44
                           58.38
                           57.67
                           35.33
       89.47               34.60
       87.88               31.04
       72.94    233.45     23.86
       67.37     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

332 :

                           75.44
                           66.57
                           58.38
                           57.67
                           48.30
       89.47               42.81
       87.88               34.60
       72.94    233.45     23.91
       67.37     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

333 :

                           91.36
                           66.57
                           57.67
                           48.30
                           42.81
       89.47               35.33
       87.88               34.60
       72.94    233.45     31.04
       67.37     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

334 :

                           91.36
                           88.40
                           78.45
       89.47               66.57
       87.88               48.30
       72.94    233.45     34.60
       67.37     98.43     18.43
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

335 :

                           95.02
                           88.40
                           75.44
                           57.67
       89.47               35.33
       87.88               26.48
       72.94    233.45     23.91
       67.37     98.43     23.86
     + 10.13   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

336 :

                           91.36
                           78.45
                           66.57
                           57.67
       89.47               42.81
       87.88               34.60
       72.94    233.45     23.91
       67.37     98.43     23.86
     + 10.13   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

337 :

       88.40
       75.44              234.65
       48.30     98.43     72.94
       35.33     89.47     57.67
       26.48     66.57     31.04
       23.86     49.30     23.91
       18.43     42.81     10.13
     + 11.55   + 34.60   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

338 :

       88.40
       75.44               95.02
       48.30     98.43     91.36
       35.33     89.47     87.88
       26.48     66.57     67.37
       23.86     49.30     57.67
       18.43     42.81     31.04
     + 11.55   + 34.60   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

339 :

       88.40
       75.44              234.65
       48.30               72.94
       35.33               57.67
       26.48               31.04
       23.86    233.45     23.91
       18.43     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

340 :

       88.40               95.02
       75.44               87.88
       48.30               78.45
       35.33               58.38
       26.48               57.67
       23.86    233.45     42.81
       18.43     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

341 :

       88.40
       75.44               95.02
       48.30               91.36
       35.33               87.88
       26.48               67.37
       23.86    233.45     57.67
       18.43     98.43     31.04
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

342 :

       95.02     98.43
       78.45     91.36    234.65
       57.67     75.44     87.88
       42.81     48.30     49.30
       23.86     31.04     34.60
       18.43     26.48     23.91
     + 11.55   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

343 :

       95.02
       78.45              234.65
       57.67     98.43     87.88
       42.81     91.36     49.30
       23.86     89.47     34.60
       18.43     75.44     23.91
     + 11.55   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

344 :

                           88.40
                           87.88
       95.02               75.44
       78.45               58.38
       57.67               48.30
       42.81               35.33
       23.86    233.45     26.48
       18.43     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

345 :

                           91.36
                           88.40
       95.02               75.44
       78.45               58.38
       57.67               35.33
       42.81               31.04
       23.86    233.45     26.48
       18.43     98.43     23.91
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

346 :

                           91.36
       95.02               89.47
       78.45               87.88
       57.67               58.38
       42.81               48.30
       23.86    233.45     31.04
       18.43     98.43     23.91
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

347 :

                           75.44
                           72.94
       95.02               66.57
       78.45               58.38
       57.67               48.30
       42.81               34.60
       23.86    233.45     31.04
       18.43     98.43     26.48
     + 11.55   + 49.30   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

348 :

       91.36
       88.40     95.02    233.45
       57.67     78.45     67.37
       31.04     75.44     49.30
       23.91     66.57     42.81
       23.86     58.38     34.60
     + 11.55   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

349 :

                           75.44
                           72.94
                           67.37
       91.36               66.57
       88.40               58.38
       57.67               34.60
       31.04               26.48
       23.91    233.45     18.43
       23.86     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

350 :

                           95.02
       91.36               89.47
       88.40               75.44
       57.67               66.57
       31.04               58.38
       23.91    233.45     35.33
       23.86     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

351 :

                           95.02
                           78.45
       91.36               75.44
       88.40               58.38
       57.67               42.81
       31.04               35.33
       23.91    233.45     26.48
       23.86     98.43     18.43
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

352 :

                 98.43
                 67.37
                 49.30
       88.40     48.30
       87.88     42.81    233.45
       58.38     31.04     78.45
       57.67     26.48     66.57
       23.91     10.13     35.33
     + 11.55   +  7.32   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

353 :

                 89.47
                 75.44
       88.40     67.37
       87.88     49.30    233.45
       58.38     42.81     78.45
       57.67     31.04     66.57
       23.91     18.43     35.33
     + 11.55   +  7.32   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

354 :

                 91.36
       88.40     89.47
       87.88     75.44    233.45
       58.38     48.30     78.45
       57.67     42.81     66.57
       23.91     26.48     35.33
     + 11.55   +  7.32   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

355 :

                 98.43
       88.40     89.47
       87.88     67.37    233.45
       58.38     49.30     78.45
       57.67     42.81     66.57
       23.91     26.48     35.33
     + 11.55   +  7.32   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

356 :

                           98.43
                           78.45
                           67.37
       88.40               66.57
       87.88    233.45     35.33
       58.38     49.30     31.04
       57.67     48.30     26.48
       23.91     42.81     23.86
     + 11.55   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

357 :

                           78.45
                           75.44
       88.40               72.94
       87.88    233.45     67.37
       58.38     49.30     66.57
       57.67     48.30     34.60
       23.91     42.81     23.86
     + 11.55   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

358 :

                 98.43
       88.40     91.36
       87.88     75.44    233.45
       58.38     48.30     78.45
       57.67     31.04     66.57
       23.91     26.48     35.33
     + 11.55   + 10.13   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

359 :

                 98.43     95.02
       88.40     91.36     89.47
       87.88     75.44     67.37
       58.38     48.30     66.57
       57.67     31.04     49.30
       23.91     26.48     35.33
     + 11.55   + 10.13   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

360 :

                           95.02
                           67.37
                           66.57
       88.40               49.30
       87.88     98.43     48.30
       58.38     91.36     35.33
       57.67     89.47     34.60
       23.91     75.44     31.04
     + 11.55   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

361 :

       88.40              233.45
       87.88     98.43     67.37
       58.38     91.36     49.30
       57.67     89.47     42.81
       23.91     75.44     34.60
     + 11.55   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

362 :

       88.40
       87.88     98.43    233.45
       58.38     91.36     78.45
       57.67     89.47     66.57
       23.91     75.44     35.33
     + 11.55   + 26.48   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

363 :

                           91.36
                           89.47
                           75.44
       88.40               67.37
       87.88               42.81
       58.38               35.33
       57.67    233.45     18.43
       23.91     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

364 :

                           78.45
                           67.37
                           66.57
                           48.30
                           42.81
       88.40               35.33
       87.88               31.04
       58.38               26.48
       57.67    233.45     23.86
       23.91     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

365 :

                           89.47
                           75.44
                           72.94
       88.40               66.57
       87.88               48.30
       58.38               35.33
       57.67    233.45     23.86
       23.91     98.43     18.43
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

366 :

                           89.47
                           78.45
                           67.37
       88.40               66.57
       87.88               42.81
       58.38               35.33
       57.67    233.45     26.48
       23.91     98.43     23.86
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

367 :

                           91.36
                           89.47
       88.40               75.44
       87.88               67.37
       58.38               42.81
       57.67    233.45     34.60
       23.91     98.43     26.48
     + 11.55   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

368 :

                           78.45
                           75.44
       88.40               72.94
       87.88               67.37
       58.38               66.57
       57.67    233.45     34.60
       23.91     98.43     23.86
     + 11.55   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

369 :

                           89.47
                           75.44
       88.40               72.94
       87.88               66.57
       58.38               48.30
       57.67    233.45     34.60
       23.91     98.43     26.48
     + 11.55   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

370 :

                           91.36
       88.40               89.47
       87.88               75.44
       58.38               72.94
       57.67    233.45     42.81
       23.91     98.43     34.60
     + 11.55   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

371 :

                           95.02
                           88.40
       87.88               78.45
       75.44               58.38
       48.30               57.67
       42.81               23.86
       35.33    233.45     18.43
       26.48     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

372 :

                           91.36
                           89.47
       87.88               67.37
       75.44               66.57
       48.30               57.67
       42.81               23.91
       35.33    233.45     23.86
       26.48     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

373 :

       87.88              234.65
       75.44               72.94
       48.30               57.67
       42.81               31.04
       35.33    233.45     23.91
       26.48     98.43     10.13
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

374 :

       87.88
       75.44              234.65
       48.30               95.02
       42.81               58.38
       35.33    233.45     23.86
       26.48     98.43     18.43
     + 11.55   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

375 :

       87.88               91.36
       75.44               89.47
       48.30               78.45
       42.81               72.94
       35.33    233.45     57.67
       26.48     98.43     23.91
     + 11.55   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

376 :

       88.40               95.02
       67.37               89.47
       48.30               75.44
       31.04               66.57
       26.48               58.38
       23.91    233.45     35.33
       23.86     98.43     10.13
     + 18.43   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

377 :

       88.40               95.02
       67.37               87.88
       48.30               78.45
       31.04               58.38
       26.48               57.67
       23.91    233.45     42.81
       23.86     98.43     10.13
     + 18.43   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

378 :

       88.40               95.02
       67.37               89.47
       48.30               75.44
       31.04               58.38
       26.48               57.67
       23.91    233.45     42.81
       23.86     98.43     11.55
     + 18.43   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

379 :

       88.40               95.02
       67.37               89.47
       48.30               75.44
       31.04               57.67
       26.48               42.81
       23.91    233.45     35.33
       23.86     98.43     34.60
     + 18.43   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

380 :

       88.40
       67.37               91.36
       48.30               87.88
       31.04               78.45
       26.48               75.44
       23.91    233.45     58.38
       23.86     98.43     34.60
     + 18.43   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

381 :

                 88.40
                 75.44    234.65
       98.43     66.57     72.94
       89.47     42.81     57.67
       49.30     35.33     31.04
       48.30     34.60     23.91
       23.86     26.48     10.13
     + 18.43   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

382 :

                 88.40
                 75.44     95.02
       98.43     66.57     91.36
       89.47     42.81     87.88
       49.30     35.33     67.37
       48.30     34.60     57.67
       23.86     26.48     31.04
     + 18.43   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

383 :

                          233.45
       98.43     95.02     66.57
       89.47     88.40     58.38
       49.30     75.44     35.33
       48.30     67.37     26.48
       23.86     31.04     10.13
     + 18.43   + 23.91   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

384 :

                          233.45
       98.43     95.02     58.38
       89.47     88.40     57.67
       49.30     75.44     42.81
       48.30     67.37     26.48
       23.86     31.04     11.55
     + 18.43   + 23.91   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

385 :

                          233.45
       98.43     95.02     57.67
       89.47     88.40     42.81
       49.30     75.44     35.33
       48.30     67.37     34.60
       23.86     31.04     26.48
     + 18.43   + 23.91   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

386 :

       91.36
       89.47     95.02    234.65
       42.81     78.45     98.43
       35.33     75.44     48.30
       26.48     66.57     34.60
       23.91     58.38     11.55
     + 18.43   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

387 :

                           95.02
                           72.94
                           67.37
       91.36               66.57
       89.47               48.30
       42.81               34.60
       35.33               23.86
       26.48    233.45     11.55
       23.91     98.43     10.13
     + 18.43   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

388 :

                           78.45
                           75.44
       91.36               67.37
       89.47               58.38
       42.81               57.67
       35.33               48.30
       26.48    233.45     34.60
       23.91     98.43     10.13
     + 18.43   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

389 :

                           95.02
       91.36               88.40
       89.47               78.45
       42.81               75.44
       35.33               48.30
       26.48    233.45     34.60
       23.91     98.43     10.13
     + 18.43   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

390 :

       98.43
       58.38     95.02
       49.30     91.36    233.45
       48.30     89.47     78.45
       31.04     87.88     66.57
       23.91     10.13     35.33
     + 18.43   +  7.32   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

391 :

                 75.44
       98.43     72.94
       89.47     66.57    234.65
       34.60     57.67     88.40
       31.04     48.30     49.30
       26.48     42.81     35.33
       23.91     10.13     18.43
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

392 :

       98.43               95.02
       89.47               88.40
       34.60    233.45     75.44
       31.04     49.30     67.37
       26.48     48.30     57.67
       23.91     42.81     35.33
     + 23.86   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

393 :

       98.43
       89.47     95.02    234.65
       34.60     78.45     88.40
       31.04     75.44     49.30
       26.48     66.57     35.33
       23.91     58.38     18.43
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

394 :

                           95.02
       98.43               91.36
       89.47               72.94
       34.60               57.67
       31.04    233.45     49.30
       26.48     87.88     42.81
       23.91     48.30     18.43
     + 23.86   + 11.55   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

395 :

       98.43               95.02
       89.47               88.40
       34.60               75.44
       31.04    233.45     67.37
       26.48     87.88     57.67
       23.91     48.30     35.33
     + 23.86   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

396 :

                 89.47
       75.44     88.40
       66.57     72.94    234.65
       57.67     42.81     98.43
       49.30     35.33     48.30
       31.04     26.48     34.60
       23.91     18.43     11.55
     + 23.86   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

397 :

       75.44               98.43
       66.57               91.36
       57.67               89.47
       49.30    233.45     72.94
       31.04     87.88     42.81
       23.91     48.30     35.33
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

398 :

                           88.40
                           78.45
                           67.37
       98.43               66.57
       87.88    233.45     57.67
       58.38     49.30     31.04
       35.33     48.30     26.48
       23.91     42.81     11.55
     + 23.86   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

399 :

       98.43              234.65
       87.88    233.45     67.37
       58.38     49.30     66.57
       35.33     48.30     31.04
       23.91     42.81     26.48
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

400 :

                 75.44
                 67.37
       95.02     66.57    234.65
       78.45     57.67     88.40
       72.94     48.30     49.30
       31.04     34.60     35.33
       26.48     23.91     18.43
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

401 :

                           98.43
       95.02               88.40
       78.45    233.45     67.37
       72.94     49.30     66.57
       31.04     48.30     58.38
       26.48     42.81     34.60
     + 23.86   +  7.32   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

402 :

                           91.36
                           89.47
                           75.44
       95.02               67.37
       78.45               42.81
       72.94    233.45     35.33
       31.04     87.88     18.43
       26.48     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

403 :

                           98.43
       95.02               88.40
       78.45               67.37
       72.94    233.45     66.57
       31.04     87.88     58.38
       26.48     48.30     34.60
     + 23.86   + 11.55   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

404 :

       95.02     98.43    234.65
       78.45     88.40     87.88
       72.94     75.44     49.30
       31.04     57.67     34.60
       26.48     42.81     23.91
     + 23.86   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

405 :

       95.02     87.88    234.65
       78.45     75.44     98.43
       72.94     67.37     48.30
       31.04     58.38     34.60
       26.48     49.30     11.55
     + 23.86   + 42.81   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

406 :

                           91.36
                           75.44
                           66.57
                           58.38
       95.02               57.67
       78.45               35.33
       72.94               23.91
       31.04    233.45     11.55
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

407 :

                           91.36
                           89.47
                           75.44
       95.02               67.37
       78.45               42.81
       72.94               35.33
       31.04    233.45     18.43
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

408 :

                           88.40
                           67.37
                           66.57
       95.02               58.38
       78.45               48.30
       72.94               42.81
       31.04    233.45     34.60
       26.48     98.43     23.91
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

409 :

                           91.36
                           89.47
                           75.44
       95.02               58.38
       78.45               48.30
       72.94               34.60
       31.04    233.45     18.43
       26.48     98.43     11.55
     + 23.86   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

410 :

                           75.44
                           66.57
                           58.38
                           57.67
       95.02               48.30
       78.45               42.81
       72.94               34.60
       31.04    233.45     23.91
       26.48     98.43     18.43
     + 23.86   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

411 :

                           89.47
                           88.40
       95.02               75.44
       78.45               58.38
       72.94               48.30
       31.04    233.45     35.33
       26.48     98.43     23.91
     + 23.86   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

412 :

                 89.47
                 75.44     98.43
       88.40     67.37     95.02
       87.88     49.30     91.36
       66.57     42.81     57.67
       34.60     31.04     48.30
       26.48     18.43     35.33
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

413 :

                 98.43
       88.40     78.45    234.65
       87.88     72.94     75.44
       66.57     57.67     49.30
       34.60     35.33     42.81
       26.48     31.04     23.91
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

414 :

                           98.43
                           95.02
       88.40               75.44
       87.88    233.45     58.38
       66.57     49.30     57.67
       34.60     48.30     31.04
       26.48     42.81     11.55
     + 23.86   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

415 :

                 98.43
       88.40     72.94    234.65
       87.88     58.38     75.44
       66.57     57.67     49.30
       34.60     48.30     42.81
       26.48     35.33     23.91
     + 23.86   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

416 :

                           95.02
                           75.44
                           58.38
                           57.67
       88.40               48.30
       87.88               42.81
       66.57               31.04
       34.60    233.45     11.55
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

417 :

                           78.45
                           72.94
                           67.37
                           58.38
       88.40               57.67
       87.88               42.81
       66.57               31.04
       34.60    233.45     11.55
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

418 :

                           89.47
                           78.45
                           72.94
                           57.67
       88.40               48.30
       87.88               31.04
       66.57               23.91
       34.60    233.45     18.43
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

419 :

                           95.02
                           91.36
                           75.44
       88.40               58.38
       87.88               57.67
       66.57               23.91
       34.60    233.45     18.43
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

420 :

                           91.36
                           89.47
                           75.44
       88.40               67.37
       87.88               42.81
       66.57               35.33
       34.60    233.45     18.43
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

421 :

                          234.65
       88.40               72.94
       87.88               57.67
       66.57               31.04
       34.60    233.45     23.91
       26.48     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

422 :

                           95.02
                           89.47
       88.40               75.44
       87.88               58.38
       66.57               57.67
       34.60    233.45     42.81
       26.48     98.43     11.55
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

423 :

                           95.02
                           75.44
                           67.37
                           57.67
       88.40               42.81
       87.88               35.33
       66.57               23.91
       34.60    233.45     18.43
       26.48     98.43     11.55
     + 23.86   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

424 :

                           89.47
                           75.44
       88.40               72.94
       87.88               58.38
       66.57               57.67
       34.60    233.45     48.30
       26.48     98.43     23.91
     + 23.86   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

425 :

                           91.36
                           89.47
       88.40               72.94
       87.88               57.67
       66.57               48.30
       34.60    233.45     35.33
       26.48     98.43     31.04
     + 23.86   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

426 :

                           91.36
                           89.47
       88.40               75.44
       87.88               48.30
       66.57               42.81
       34.60    233.45     35.33
       26.48     98.43     31.04
     + 23.86   + 49.30   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

427 :

                           91.36
                           89.47
       72.94               75.44
       58.38               67.37
       57.67               42.81
       49.30    233.45     35.33
       34.60     87.88     18.43
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

428 :

       72.94               98.43
       58.38               91.36
       57.67               78.45
       49.30    233.45     67.37
       34.60     87.88     42.81
       31.04     48.30     35.33
     + 23.86   + 11.55   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

429 :

                           91.36
                           87.88
                           75.44
       89.47               72.94
       67.37    233.45     35.33
       58.38     49.30     34.60
       57.67     48.30     18.43
       31.04     42.81     11.55
     + 23.86   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

430 :

                           98.43
                           95.02
       89.47               88.40
       67.37    233.45     75.44
       58.38     49.30     26.48
       57.67     48.30     23.91
       31.04     42.81     18.43
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

431 :

                 95.02
       89.47     75.44    234.65
       67.37     72.94     88.40
       58.38     66.57     49.30
       57.67     34.60     35.33
       31.04     26.48     18.43
     + 23.86   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

432 :

                           91.36
                           75.44
                           72.94
                           49.30
       89.47               42.81
       67.37               35.33
       58.38    233.45     34.60
       57.67     87.88     18.43
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

433 :

                           98.43
                           88.40
                           78.45
       89.47               66.57
       67.37               35.33
       58.38    233.45     34.60
       57.67     87.88     18.43
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

434 :

                           98.43
                           88.40
       89.47               75.44
       67.37               72.94
       58.38    233.45     66.57
       57.67     87.88     18.43
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

435 :

                           98.43
                           78.45
                           75.44
       89.47               66.57
       67.37               42.81
       58.38    233.45     34.60
       57.67     87.88     23.91
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

436 :

                           88.40
                           78.45
                           75.44
                           49.30
       89.47               35.33
       67.37               34.60
       58.38    233.45     26.48
       57.67     87.88     23.91
       31.04     48.30     18.43
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

437 :

       89.47               98.43
       67.37               95.02
       58.38    233.45     91.36
       57.67     87.88     75.44
       31.04     48.30     42.81
     + 23.86   + 11.55   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

438 :

                           95.02
                           78.45
                           66.57
                           48.30
                           35.33
       89.47               34.60
       67.37               26.48
       58.38               23.91
       57.67    233.45     11.55
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

439 :

                           95.02
                           75.44
                           72.94
                           66.57
       89.47               48.30
       67.37               26.48
       58.38               23.91
       57.67    233.45     11.55
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

440 :

                           95.02
                           88.40
       89.47               78.45
       67.37               75.44
       58.38               48.30
       57.67    233.45     34.60
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

441 :

                           95.02
                           88.40
                           75.44
                           48.30
       89.47               42.81
       67.37               26.48
       58.38               23.91
       57.67    233.45     18.43
       31.04     98.43     11.55
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

442 :

                           91.36
                           87.88
                           75.44
       89.47               72.94
       67.37               35.33
       58.38               34.60
       57.67    233.45     18.43
       31.04     98.43     11.55
     + 23.86   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

443 :

                           88.40
                           78.45
                           72.94
       89.47               66.57
       67.37               42.81
       58.38               34.60
       57.67    233.45     23.91
       31.04     98.43     18.43
     + 23.86   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

444 :

                           91.36
                           88.40
       89.47               78.45
       67.37               66.57
       58.38               48.30
       57.67    233.45     34.60
       31.04     98.43     18.43
     + 23.86   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

445 :

                 78.45
                 66.57
                 58.38     98.43
                 57.67     87.88
       95.02     48.30     75.44
       89.47     42.81     67.37
       88.40     11.55     49.30
       31.04     10.13     35.33
     + 23.86   +  7.32   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

446 :

                 98.43
                 78.45
                 67.37    234.65
       95.02     57.67     75.44
       89.47     35.33     49.30
       88.40     26.48     42.81
       31.04     10.13     23.91
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

447 :

                 91.36
                 78.45    234.65
       95.02     75.44     98.43
       89.47     67.37     48.30
       88.40     42.81     34.60
       31.04     18.43     11.55
     + 23.86   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

448 :

                           91.36
                           87.88
                           75.44
                           72.94
       95.02    233.45     35.33
       89.47     49.30     34.60
       88.40     48.30     18.43
       31.04     42.81     11.55
     + 23.86   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

449 :

                           98.43
                           75.44
                           67.37
                           58.38
       95.02    233.45     57.67
       89.47     49.30     26.48
       88.40     48.30     23.91
       31.04     42.81     18.43
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

450 :

                           91.36
                           87.88
       95.02    233.45     78.45
       89.47     49.30     75.44
       88.40     48.30     58.38
       31.04     42.81     34.60
     + 23.86   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

451 :

                           98.43
                           75.44
                           67.37
                           57.67
       95.02    233.45     35.33
       89.47     49.30     34.60
       88.40     48.30     26.48
       31.04     42.81     23.91
     + 23.86   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

452 :

                 98.43
                 78.45    234.65
       95.02     66.57     87.88
       89.47     58.38     49.30
       88.40     57.67     34.60
       31.04     11.55     23.91
     + 23.86   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

453 :

                           87.88
                           75.44
                 98.43     67.37
                 78.45     49.30
       95.02     66.57     48.30
       89.47     58.38     42.81
       88.40     57.67     35.33
       31.04     11.55     23.91
     + 23.86   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

454 :

                           87.88
                 98.43     75.44
                 78.45     72.94
       95.02     66.57     67.37
       89.47     58.38     49.30
       88.40     57.67     42.81
       31.04     11.55     34.60
     + 23.86   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

455 :

                 91.36
                 87.88    233.45
                 66.57     58.38
       95.02     48.30     57.67
       89.47     34.60     42.81
       88.40     23.91     26.48
       31.04     18.43     11.55
     + 23.86   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

456 :

                 91.36
                 87.88     98.43
                 66.57     78.45
       95.02     48.30     75.44
       89.47     34.60     58.38
       88.40     23.91     57.67
       31.04     18.43     42.81
     + 23.86   + 10.13   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

457 :

                 98.43
                 78.45    234.65
       95.02     66.57     75.44
       89.47     57.67     49.30
       88.40     35.33     42.81
       31.04     34.60     23.91
     + 23.86   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

458 :

                 98.43
                 72.94    234.65
       95.02     58.38     87.88
       89.47     57.67     49.30
       88.40     48.30     34.60
       31.04     35.33     23.91
     + 23.86   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

459 :

                 98.43
                 72.94    234.65
       95.02     58.38     75.44
       89.47     57.67     49.30
       88.40     48.30     42.81
       31.04     35.33     23.91
     + 23.86   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

460 :

                           87.88
                           78.45
                           67.37
                 98.43     58.38
       95.02     75.44     49.30
       89.47     72.94     42.81
       88.40     66.57     34.60
       31.04     57.67     11.55
     + 23.86   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

461 :

                 98.43    234.65
       95.02     75.44     87.88
       89.47     72.94     49.30
       88.40     66.57     34.60
       31.04     57.67     23.91
     + 23.86   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

462 :

                 91.36
                 78.45    234.65
       95.02     67.37     87.88
       89.47     57.67     49.30
       88.40     48.30     34.60
       31.04     26.48     23.91
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

463 :

                           91.36
                           66.57
                           58.38
                           57.67
                           42.81
                           34.60
       95.02               26.48
       89.47    233.45     23.91
       88.40     87.88     18.43
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

464 :

                           75.44
                           72.94
                           67.37
                           66.57
                           58.38
       95.02               34.60
       89.47    233.45     26.48
       88.40     87.88     18.43
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

465 :

                           91.36
                           75.44
                           72.94
                           49.30
                           42.81
       95.02               35.33
       89.47    233.45     34.60
       88.40     87.88     18.43
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

466 :

                           98.43
                           78.45
                           75.44
                           66.57
       95.02               42.81
       89.47    233.45     34.60
       88.40     87.88     23.91
       31.04     48.30     10.13
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

467 :

                           91.36
                           78.45
                           75.44
       95.02               58.38
       89.47    233.45     49.30
       88.40     87.88     42.81
       31.04     48.30     34.60
     + 23.86   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

468 :

                           98.43
                           75.44
                           72.94
                           57.67
       95.02               42.81
       89.47    233.45     35.33
       88.40     87.88     26.48
       31.04     48.30     18.43
     + 23.86   + 11.55   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

469 :

                           98.43
                           75.44
                           67.37
                           57.67
       95.02               35.33
       89.47    233.45     34.60
       88.40     87.88     26.48
       31.04     48.30     23.91
     + 23.86   + 11.55   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

470 :

                           98.43
                           91.36
       95.02               78.45
       89.47    233.45     67.37
       88.40     87.88     42.81
       31.04     48.30     35.33
     + 23.86   + 11.55   + 23.91
     =======   =======   =======
      327.79    381.18    437.66

471 :

                           98.43
                           78.45
       95.02               75.44
       89.47    233.45     58.38
       88.40     87.88     57.67
       31.04     48.30     42.81
     + 23.86   + 11.55   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

472 :

                 98.43    233.45
                 75.44     66.57
       95.02     67.37     58.38
       89.47     49.30     35.33
       88.40     48.30     26.48
       31.04     23.91     10.13
     + 23.86   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

473 :

                 98.43    233.45
                 75.44     58.38
       95.02     67.37     57.67
       89.47     49.30     42.81
       88.40     48.30     26.48
       31.04     23.91     11.55
     + 23.86   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

474 :

                 98.43    233.45
                 75.44     57.67
       95.02     67.37     42.81
       89.47     49.30     35.33
       88.40     48.30     34.60
       31.04     23.91     26.48
     + 23.86   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

475 :

                           98.43
                           78.45
                           66.57
                           57.67
                 87.88     48.30
       95.02     75.44     35.33
       89.47     67.37     23.91
       88.40     58.38     11.55
       31.04     49.30     10.13
     + 23.86   + 42.81   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

476 :

                           98.43
                           78.45
                           72.94
                 87.88     66.57
       95.02     75.44     57.67
       89.47     67.37     34.60
       88.40     58.38     11.55
       31.04     49.30     10.13
     + 23.86   + 42.81   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

477 :

                 87.88    234.65
       95.02     75.44     98.43
       89.47     67.37     48.30
       88.40     58.38     34.60
       31.04     49.30     11.55
     + 23.86   + 42.81   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

478 :

                           91.36
                           75.44
                           66.57
                           58.38
                           57.67
       95.02               35.33
       89.47               23.91
       88.40    233.45     11.55
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

479 :

                           91.36
                           66.57
                           58.38
                           57.67
                           42.81
                           34.60
       95.02               26.48
       89.47               23.91
       88.40    233.45     18.43
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

480 :

                           75.44
                           67.37
                           66.57
                           58.38
                           48.30
                           35.33
       95.02               26.48
       89.47               23.91
       88.40    233.45     18.43
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

481 :

                           75.44
                           72.94
                           67.37
                           66.57
                           58.38
       95.02               34.60
       89.47               26.48
       88.40    233.45     18.43
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

482 :

                           78.45
                           75.44
                           67.37
                           58.38
       95.02               57.67
       89.47               48.30
       88.40    233.45     34.60
       31.04     98.43     10.13
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

483 :

                           75.44
                           67.37
                           58.38
                           57.67
                           48.30
                           42.81
       95.02               26.48
       89.47               23.91
       88.40    233.45     18.43
       31.04     98.43     11.55
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

484 :

                           75.44
                           67.37
                           57.67
                           48.30
                           42.81
                           35.33
       95.02               34.60
       89.47               26.48
       88.40    233.45     23.91
       31.04     98.43     18.43
     + 23.86   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

485 :

                           91.36
                           87.88
                           75.44
                           72.94
       95.02               35.33
       89.47               34.60
       88.40    233.45     18.43
       31.04     98.43     11.55
     + 23.86   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

486 :

                           87.88
                           78.45
                           72.94
       95.02               58.38
       89.47               57.67
       88.40    233.45     48.30
       31.04     98.43     23.91
     + 23.86   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

487 :

                           75.44
                           66.57
                           58.38
                           57.67
                           48.30
       95.02               42.81
       89.47               34.60
       88.40    233.45     23.91
       31.04     98.43     18.43
     + 23.86   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

488 :

                           91.36
                           87.88
       95.02               78.45
       89.47               75.44
       88.40    233.45     58.38
       31.04     98.43     34.60
     + 23.86   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

489 :

                           78.45
                           75.44
                           72.94
                           58.38
       95.02               48.30
       89.47               35.33
       88.40    233.45     26.48
       31.04     98.43     23.91
     + 23.86   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

490 :

                           91.36
                           87.88
                           72.94
       95.02               57.67
       89.47               48.30
       88.40    233.45     34.60
       31.04     98.43     26.48
     + 23.86   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

491 :

                           91.36
                           87.88
       95.02               72.94
       89.47               66.57
       88.40    233.45     57.67
       31.04     98.43     42.81
     + 23.86   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

492 :

                 98.43
       87.88     88.40
       67.37     58.38
       48.30     57.67    233.45
       42.81     49.30     78.45
       31.04     11.55     66.57
       26.48     10.13     35.33
     + 23.91   +  7.32   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

493 :

       87.88
       67.37     95.02    234.65
       48.30     78.45     88.40
       42.81     75.44     49.30
       31.04     66.57     35.33
       26.48     58.38     18.43
     + 23.91   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

494 :

       87.88     98.43
       67.37     78.45    233.45
       48.30     66.57     88.40
       42.81     58.38     49.30
       31.04     57.67     35.33
       26.48     11.55     23.86
     + 23.91   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

495 :

       87.88     98.43
       67.37     78.45    234.65
       48.30     66.57     95.02
       42.81     57.67     58.38
       31.04     35.33     23.86
       26.48     34.60     18.43
     + 23.91   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

496 :

       87.88
       67.37     98.43    234.65
       48.30     75.44     95.02
       42.81     72.94     58.38
       31.04     66.57     23.86
       26.48     57.67     18.43
     + 23.91   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

497 :

       87.88
       67.37     98.43    233.45
       48.30     75.44     88.40
       42.81     72.94     49.30
       31.04     66.57     35.33
       26.48     57.67     23.86
     + 23.91   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

498 :

       87.88
       67.37     98.43    234.65
       48.30     75.44     88.40
       42.81     72.94     49.30
       31.04     66.57     35.33
       26.48     57.67     18.43
     + 23.91   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

499 :

                           88.40
                           78.45
                           66.57
       87.88               58.38
       67.37               57.67
       48.30               35.33
       42.81               23.86
       31.04    233.45     11.55
       26.48     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

500 :

                           95.02
                           88.40
       87.88               78.45
       67.37               58.38
       48.30               57.67
       42.81               23.86
       31.04    233.45     18.43
       26.48     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

501 :

                           88.40
                           75.44
       87.88               72.94
       67.37               66.57
       48.30               57.67
       42.81               35.33
       31.04    233.45     23.86
       26.48     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

502 :

                           95.02
       87.88               89.47
       67.37               75.44
       48.30               66.57
       42.81               58.38
       31.04    233.45     35.33
       26.48     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

503 :

       87.88              234.65
       67.37               66.57
       48.30               58.38
       42.81               35.33
       31.04    233.45     23.86
       26.48     98.43     11.55
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

504 :

       87.88
       67.37              234.65
       48.30               95.02
       42.81               58.38
       31.04    233.45     23.86
       26.48     98.43     18.43
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

505 :

       87.88               89.47
       67.37               88.40
       48.30               75.44
       42.81               72.94
       31.04    233.45     58.38
       26.48     98.43     34.60
     + 23.91   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

506 :

       78.45
       58.38     95.02    234.65
       57.67     91.36     88.40
       48.30     89.47     49.30
       34.60     87.88     35.33
       26.48     10.13     18.43
     + 23.91   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

507 :

       78.45     89.47
       58.38     87.88    233.45
       57.67     75.44     88.40
       48.30     67.37     49.30
       34.60     31.04     35.33
       26.48     18.43     23.86
     + 23.91   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

508 :

                           95.02
                           89.47
                           67.37
       78.45               66.57
       58.38               35.33
       57.67               31.04
       48.30               23.86
       34.60    233.45     11.55
       26.48     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

509 :

                           91.36
                           89.47
       78.45               75.44
       58.38               67.37
       57.67               42.81
       48.30               35.33
       34.60    233.45     18.43
       26.48     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

510 :

                           95.02
                           75.44
       78.45               72.94
       58.38               66.57
       57.67               42.81
       48.30               31.04
       34.60    233.45     23.86
       26.48     98.43     18.43
     + 23.91   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

511 :

       88.40              234.65
       87.88    233.45     67.37
       57.67     49.30     66.57
       35.33     48.30     31.04
       34.60     42.81     26.48
     + 23.91   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

512 :

                 98.43
       88.40     91.36    234.65
       87.88     75.44     95.02
       57.67     48.30     58.38
       35.33     31.04     23.86
       34.60     26.48     18.43
     + 23.91   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

513 :

                 98.43     95.02
       88.40     91.36     89.47
       87.88     75.44     67.37
       57.67     48.30     66.57
       35.33     31.04     58.38
       34.60     26.48     49.30
     + 23.91   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

514 :

       88.40              234.65
       87.88     98.43     95.02
       57.67     91.36     58.38
       35.33     89.47     23.86
       34.60     75.44     18.43
     + 23.91   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

515 :

                           95.02
                           67.37
                           66.57
       88.40               58.38
       87.88     98.43     49.30
       57.67     91.36     48.30
       35.33     89.47     31.04
       34.60     75.44     11.55
     + 23.91   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

516 :

       88.40              234.65
       87.88               95.02
       57.67               58.38
       35.33    233.45     23.86
       34.60     98.43     18.43
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

517 :

                           95.02
                           75.44
                           67.37
                           66.57
       88.40               42.81
       87.88               26.48
       57.67               23.86
       35.33    233.45     18.43
       34.60     98.43     11.55
     + 23.91   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

518 :

                           95.02
                           75.44
                           72.94
       88.40               66.57
       87.88               42.81
       57.67               31.04
       35.33    233.45     23.86
       34.60     98.43     18.43
     + 23.91   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

519 :

                           89.47
                           78.45
                           67.37
       88.40               66.57
       87.88               48.30
       57.67               31.04
       35.33    233.45     26.48
       34.60     98.43     18.43
     + 23.91   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

520 :

       88.40              234.65
       87.88               67.37
       57.67               66.57
       35.33    233.45     31.04
       34.60     98.43     26.48
     + 23.91   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

521 :

                           95.02
                           75.44
       88.40               67.37
       87.88               58.38
       57.67               48.30
       35.33    233.45     42.81
       34.60     98.43     26.48
     + 23.91   + 49.30   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

522 :

       78.45     95.02    234.65
       66.57     91.36     88.40
       58.38     89.47     49.30
       57.67     87.88     35.33
       42.81     10.13     18.43
     + 23.91   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

523 :

                 98.43
                 87.88
       78.45     67.37    233.45
       66.57     48.30     88.40
       58.38     31.04     49.30
       57.67     26.48     35.33
       42.81     11.55     23.86
     + 23.91   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

524 :

                 98.43
       78.45     91.36    233.45
       66.57     75.44     88.40
       58.38     48.30     49.30
       57.67     31.04     35.33
       42.81     26.48     23.86
     + 23.91   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

525 :

                 98.43
       78.45     91.36    234.65
       66.57     75.44     88.40
       58.38     48.30     49.30
       57.67     31.04     35.33
       42.81     26.48     18.43
     + 23.91   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

526 :

                 98.43     95.02
       78.45     91.36     89.47
       66.57     75.44     88.40
       58.38     48.30     87.88
       57.67     31.04     34.60
       42.81     26.48     23.86
     + 23.91   + 10.13   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

527 :

                 89.47
       78.45     87.88    233.45
       66.57     75.44     88.40
       58.38     67.37     49.30
       57.67     31.04     35.33
       42.81     18.43     23.86
     + 23.91   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

528 :

       78.45     98.43    233.45
       66.57     89.47     88.40
       58.38     87.88     49.30
       57.67     67.37     35.33
       42.81     26.48     23.86
     + 23.91   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

529 :

                           98.43
                           88.40
                           67.37
                           49.30
       78.45               35.33
       66.57               31.04
       58.38    233.45     26.48
       57.67     87.88     23.86
       42.81     48.30     10.13
     + 23.91   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

530 :

                           98.43
                           89.47
                           75.44
       78.45               67.37
       66.57               34.60
       58.38    233.45     31.04
       57.67     87.88     23.86
       42.81     48.30     10.13
     + 23.91   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

531 :

                           88.40
                           75.44
                           72.94
       78.45               67.37
       66.57               49.30
       58.38    233.45     34.60
       57.67     87.88     23.86
       42.81     48.30     18.43
     + 23.91   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

532 :

                           91.36
                           89.47
       78.45               88.40
       66.57               75.44
       58.38    233.45     35.33
       57.67     87.88     26.48
       42.81     48.30     23.86
     + 23.91   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

533 :

       78.45              233.45
       66.57     98.43     88.40
       58.38     91.36     49.30
       57.67     89.47     35.33
       42.81     75.44     23.86
     + 23.91   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

534 :

                           95.02
                           88.40
                           87.88
       78.45               48.30
       66.57     98.43     34.60
       58.38     91.36     31.04
       57.67     89.47     23.86
       42.81     75.44     18.43
     + 23.91   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

535 :

       78.45              234.65
       66.57     98.43     88.40
       58.38     91.36     49.30
       57.67     89.47     35.33
       42.81     75.44     18.43
     + 23.91   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

536 :

                           88.40
                           87.88
                           67.37
                           48.30
                           35.33
       78.45               31.04
       66.57               26.48
       58.38               23.86
       57.67    233.45     11.55
       42.81     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

537 :

                           91.36
                           88.40
                           75.44
                           48.30
       78.45               35.33
       66.57               31.04
       58.38               26.48
       57.67    233.45     23.86
       42.81     98.43     10.13
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

538 :

                           89.47
                           88.40
                           87.88
       78.45               67.37
       66.57               35.33
       58.38               26.48
       57.67    233.45     23.86
       42.81     98.43     11.55
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

539 :

                           91.36
                           89.47
       78.45               88.40
       66.57               75.44
       58.38               35.33
       57.67    233.45     26.48
       42.81     98.43     23.86
     + 23.91   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

540 :

                           91.36
                           87.88
                           75.44
       78.45               72.94
       66.57               35.33
       58.38               34.60
       57.67    233.45     18.43
       42.81     98.43     11.55
     + 23.91   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

541 :

                           95.02
                           88.40
                           87.88
       78.45               48.30
       66.57               34.60
       58.38               31.04
       57.67    233.45     23.86
       42.81     98.43     18.43
     + 23.91   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

542 :

                           95.02
                           75.44
                           72.94
                           48.30
       78.45               34.60
       66.57               31.04
       58.38               26.48
       57.67    233.45     23.86
       42.81     98.43     18.43
     + 23.91   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

543 :

                           89.47
                           88.40
                           72.94
       78.45               67.37
       66.57               34.60
       58.38               31.04
       57.67    233.45     23.86
       42.81     98.43     18.43
     + 23.91   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

544 :

                           95.02
       78.45               89.47
       66.57               88.40
       58.38               87.88
       57.67    233.45     34.60
       42.81     98.43     23.86
     + 23.91   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

545 :

                 98.43
                 78.45     88.40
       91.36     75.44     87.88
       72.94     58.38     67.37
       57.67     34.60     66.57
       48.30     18.43     49.30
       31.04     10.13     42.81
     + 26.48   +  7.32   + 35.33
     =======   =======   =======
      327.79    381.18    437.66

546 :

       91.36     95.02    233.45
       72.94     78.45     67.37
       57.67     75.44     49.30
       48.30     66.57     42.81
       31.04     58.38     34.60
     + 26.48   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

547 :

       91.36     95.02    234.65
       72.94     78.45     88.40
       57.67     75.44     49.30
       48.30     66.57     35.33
       31.04     58.38     18.43
     + 26.48   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

548 :

                 89.47
       91.36     88.40    234.65
       72.94     67.37     95.02
       57.67     66.57     58.38
       48.30     35.33     23.86
       31.04     23.91     18.43
     + 26.48   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

549 :

       91.36     98.43    234.65
       72.94     89.47     95.02
       57.67     66.57     58.38
       48.30     49.30     23.86
       31.04     42.81     18.43
     + 26.48   + 34.60   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

550 :

                           98.43
                           88.40
                           78.45
       91.36     87.88     66.57
       72.94     75.44     35.33
       57.67     67.37     34.60
       48.30     58.38     18.43
       31.04     49.30     10.13
     + 26.48   + 42.81   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

551 :

       91.36     87.88
       72.94     75.44    233.45
       57.67     67.37     78.45
       48.30     58.38     66.57
       31.04     49.30     35.33
     + 26.48   + 42.81   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

552 :

                           95.02
                           89.47
       91.36               75.44
       72.94               66.57
       57.67               58.38
       48.30    233.45     35.33
       31.04     98.43     10.13
     + 26.48   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

553 :

                          234.65
       91.36               66.57
       72.94               58.38
       57.67               35.33
       48.30    233.45     23.86
       31.04     98.43     11.55
     + 26.48   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

554 :

       91.36              234.65
       72.94               95.02
       57.67               58.38
       48.30    233.45     23.86
       31.04     98.43     18.43
     + 26.48   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

555 :

                           95.02
                           89.47
                           87.88
                           42.81
       91.36               34.60
       72.94               23.91
       57.67               23.86
       48.30    233.45     18.43
       31.04     98.43     11.55
     + 26.48   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

556 :

                           89.47
                           88.40
       91.36               87.88
       72.94               66.57
       57.67               35.33
       48.30    233.45     34.60
       31.04     98.43     23.86
     + 26.48   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

557 :

                           95.02
       91.36               89.47
       72.94               88.40
       57.67               87.88
       48.30    233.45     34.60
       31.04     98.43     23.86
     + 26.48   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

558 :

                 91.36
                 67.37     95.02
       98.43     66.57     89.47
       75.44     57.67     88.40
       49.30     48.30     87.88
       42.81     31.04     34.60
       35.33     11.55     23.86
     + 26.48   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

559 :

                 89.47
                 88.40    234.65
       98.43     66.57     72.94
       75.44     48.30     57.67
       49.30     34.60     31.04
       42.81     23.86     23.91
       35.33     18.43     10.13
     + 26.48   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

560 :

                 89.47
                 88.40     95.02
       98.43     66.57     91.36
       75.44     48.30     87.88
       49.30     34.60     67.37
       42.81     23.86     57.67
       35.33     18.43     31.04
     + 26.48   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

561 :

                           95.02
                           88.40
                           78.45
       98.43               58.38
       75.44               57.67
       49.30    233.45     23.86
       42.81     87.88     18.43
       35.33     48.30     10.13
     + 26.48   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

562 :

                           91.36
                           89.47
                           67.37
       98.43               66.57
       75.44               57.67
       49.30    233.45     23.91
       42.81     87.88     23.86
       35.33     48.30     10.13
     + 26.48   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

563 :

                          234.65
       98.43               72.94
       75.44               57.67
       49.30    233.45     31.04
       42.81     87.88     23.91
       35.33     48.30     10.13
     + 26.48   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

564 :

       98.43              234.65
       75.44               95.02
       49.30    233.45     58.38
       42.81     87.88     23.86
       35.33     48.30     18.43
     + 26.48   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

565 :

                           91.36
       98.43               89.47
       75.44               78.45
       49.30    233.45     72.94
       42.81     87.88     57.67
       35.33     48.30     23.91
     + 26.48   + 11.55   + 23.86
     =======   =======   =======
      327.79    381.18    437.66

566 :

                 98.43     89.47
       87.88     95.02     88.40
       66.57     91.36     75.44
       57.67     67.37     72.94
       49.30     11.55     58.38
       35.33     10.13     34.60
     + 31.04   +  7.32   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

567 :

                 98.43
                 78.45     91.36
       87.88     75.44     88.40
       66.57     58.38     72.94
       57.67     34.60     67.37
       49.30     18.43     48.30
       35.33     10.13     42.81
     + 31.04   +  7.32   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

568 :

                 89.47
                 88.40
                 67.37
       87.88     48.30     98.43
       66.57     26.48     95.02
       57.67     23.86     91.36
       49.30     18.43     75.44
       35.33     11.55     42.81
     + 31.04   +  7.32   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

569 :

                 91.36
       87.88     78.45    234.65
       66.57     75.44     98.43
       57.67     67.37     48.30
       49.30     42.81     34.60
       35.33     18.43     11.55
     + 31.04   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

570 :

                           98.43
                 91.36     88.40
       87.88     78.45     72.94
       66.57     75.44     58.38
       57.67     67.37     48.30
       49.30     42.81     34.60
       35.33     18.43     26.48
     + 31.04   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

571 :

                           98.43
                 91.36     95.02
       87.88     89.47     88.40
       66.57     75.44     67.37
       57.67     48.30     34.60
       49.30     42.81     23.86
       35.33     26.48     18.43
     + 31.04   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

572 :

                           98.43
                 95.02     91.36
       87.88     88.40     89.47
       66.57     75.44     67.37
       57.67     42.81     48.30
       49.30     34.60     23.86
       35.33     26.48     11.55
     + 31.04   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

573 :

                           95.02
                           88.40
                           67.37
                           48.30
       87.88               42.81
       66.57     98.43     34.60
       57.67     91.36     23.86
       49.30     89.47     18.43
       35.33     75.44     11.55
     + 31.04   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

574 :

       87.88              234.65
       66.57     98.43     95.02
       57.67     91.36     58.38
       49.30     89.47     23.86
       35.33     75.44     18.43
     + 31.04   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

575 :

                           95.02
                           88.40
                           67.37
       87.88               58.38
       66.57     98.43     48.30
       57.67     91.36     34.60
       49.30     89.47     23.91
       35.33     75.44     11.55
     + 31.04   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

576 :

                 98.43     91.36
                 78.45     88.40
                 75.44     72.94
       87.88     58.38     49.30
       67.37     34.60     42.81
       66.57     18.43     35.33
       57.67     10.13     31.04
     + 48.30   +  7.32   + 26.48
     =======   =======   =======
      327.79    381.18    437.66

577 :

                 89.47
                 88.40
                 49.30
                 35.33
                 31.04     98.43
       87.88     26.48     95.02
       67.37     23.86     91.36
       66.57     18.43     75.44
       57.67     11.55     42.81
     + 48.30   +  7.32   + 34.60
     =======   =======   =======
      327.79    381.18    437.66

578 :

                           98.43
                           78.45
                 95.02     75.44
                 89.47     49.30
       87.88     88.40     42.81
       67.37     58.38     35.33
       66.57     31.04     23.91
       57.67     11.55     23.86
     + 48.30   +  7.32   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

579 :

                 95.02
                 78.45
                 72.94    234.65
       87.88     35.33     75.44
       67.37     34.60     49.30
       66.57     31.04     42.81
       57.67     26.48     23.91
     + 48.30   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

580 :

                           98.43
                           91.36
                 95.02     75.44
                 89.47     49.30
       87.88     88.40     42.81
       67.37     35.33     26.48
       66.57     34.60     23.86
       57.67     31.04     18.43
     + 48.30   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

581 :

                 95.02
                 89.47    234.65
       87.88     88.40     75.44
       67.37     35.33     49.30
       66.57     34.60     42.81
       57.67     31.04     23.91
     + 48.30   +  7.32   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

582 :

                 98.43
                 78.45
                 58.38
                 42.81    233.45
       87.88     31.04     88.40
       67.37     26.48     49.30
       66.57     23.91     35.33
       57.67     11.55     23.86
     + 48.30   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

583 :

                 98.43     91.36
                 88.40     78.45
                 72.94     75.44
       87.88     35.33     58.38
       67.37     31.04     49.30
       66.57     26.48     42.81
       57.67     18.43     34.60
     + 48.30   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

584 :

                 98.43
                 88.40
                 72.94    234.65
       87.88     35.33     75.44
       67.37     31.04     49.30
       66.57     26.48     42.81
       57.67     18.43     23.91
     + 48.30   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

585 :

                 91.36     98.43
                 75.44     88.40
                 58.38     78.45
       87.88     49.30     72.94
       67.37     42.81     34.60
       66.57     35.33     31.04
       57.67     18.43     26.48
     + 48.30   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

586 :

                 98.43
                 78.45
                 42.81
                 35.33    234.65
       87.88     34.60     95.02
       67.37     31.04     58.38
       66.57     26.48     23.86
       57.67     23.91     18.43
     + 48.30   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

587 :

                 98.43
                 75.44
                 72.94    234.65
       87.88     42.81     95.02
       67.37     31.04     58.38
       66.57     26.48     23.86
       57.67     23.91     18.43
     + 48.30   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

588 :

                 98.43
                 75.44
                 72.94    233.45
       87.88     42.81     88.40
       67.37     31.04     49.30
       66.57     26.48     35.33
       57.67     23.91     23.86
     + 48.30   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

589 :

                 98.43
                 75.44
                 72.94    234.65
       87.88     42.81     88.40
       67.37     31.04     49.30
       66.57     26.48     35.33
       57.67     23.91     18.43
     + 48.30   + 10.13   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

590 :

                           98.43
                           89.47
                 95.02     75.44
                 88.40     58.38
       87.88     78.45     49.30
       67.37     42.81     23.91
       66.57     35.33     23.86
       57.67     31.04     11.55
     + 48.30   + 10.13   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

591 :

                 98.43     91.36
                 95.02     89.47
                 49.30     88.40
       87.88     42.81     75.44
       67.37     34.60     35.33
       66.57     31.04     26.48
       57.67     18.43     23.86
     + 48.30   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

592 :

                 91.36
                 78.45
                 72.94    233.45
       87.88     42.81     88.40
       67.37     34.60     49.30
       66.57     31.04     35.33
       57.67     18.43     23.86
     + 48.30   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

593 :

                 95.02     98.43
                 88.40     91.36
       87.88     78.45     89.47
       67.37     49.30     72.94
       66.57     34.60     42.81
       57.67     23.86     35.33
     + 48.30   + 11.55   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

594 :

                           98.43
                 95.02     89.47
                 91.36     88.40
       87.88     75.44     58.38
       67.37     49.30     35.33
       66.57     34.60     31.04
       57.67     23.91     26.48
     + 48.30   + 11.55   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

595 :

                           98.43
                           91.36
                 95.02     89.47
                 88.40     49.30
       87.88     75.44     35.33
       67.37     42.81     31.04
       66.57     34.60     23.86
       57.67     26.48     11.55
     + 48.30   + 18.43   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

596 :

                          233.45
                           49.30
                           42.81
                 98.43     35.33
       87.88     88.40     23.91
       67.37     78.45     23.86
       66.57     58.38     11.55
       57.67     31.04     10.13
     + 48.30   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

597 :

                           91.36
                           75.44
                           72.94
                           49.30
                 98.43     42.81
       87.88     88.40     35.33
       67.37     78.45     34.60
       66.57     58.38     18.43
       57.67     31.04     10.13
     + 48.30   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

598 :

                 98.43    234.65
       87.88     88.40     75.44
       67.37     78.45     49.30
       66.57     58.38     42.81
       57.67     31.04     23.91
     + 48.30   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

599 :

                          233.45
                           49.30
                           42.81
                           35.33
       87.88     98.43     23.91
       67.37     91.36     23.86
       66.57     89.47     11.55
       57.67     75.44     10.13
     + 48.30   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

600 :

                           95.02
                           88.40
                           78.45
                           72.94
       87.88     98.43     42.81
       67.37     91.36     31.04
       66.57     89.47     11.55
       57.67     75.44     10.13
     + 48.30   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

601 :

                           95.02
                           88.40
                           49.30
                           42.81
                           35.33
                           34.60
       87.88     98.43     31.04
       67.37     91.36     23.86
       66.57     89.47     18.43
       57.67     75.44     11.55
     + 48.30   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

602 :

                          234.65
       87.88     98.43     95.02
       67.37     91.36     58.38
       66.57     89.47     23.86
       57.67     75.44     18.43
     + 48.30   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

603 :

                          233.45
       87.88     98.43     88.40
       67.37     91.36     49.30
       66.57     89.47     35.33
       57.67     75.44     23.86
     + 48.30   + 26.48   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

604 :

                           95.02
                           88.40
                           58.38
                           49.30
                           35.33
       87.88     98.43     34.60
       67.37     91.36     31.04
       66.57     89.47     23.91
       57.67     75.44     11.55
     + 48.30   + 26.48   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

605 :

                          234.65
       87.88     98.43     88.40
       67.37     91.36     49.30
       66.57     89.47     35.33
       57.67     75.44     18.43
     + 48.30   + 26.48   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

606 :

                           88.40
                           78.45
                           58.38
                           42.81
                           35.33
                           31.04
                           26.48
       87.88               23.91
       67.37               23.86
       66.57    233.45     11.55
       57.67     98.43     10.13
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

607 :

                           91.36
                           89.47
                           75.44
                           42.81
                           35.33
                           26.48
       87.88               23.91
       67.37               23.86
       66.57    233.45     11.55
       57.67     98.43     10.13
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

608 :

                           95.02
                           89.47
                           72.94
                           58.38
                           35.33
       87.88               31.04
       67.37               26.48
       66.57    233.45     11.55
       57.67     98.43     10.13
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

609 :

                           95.02
                           88.40
                           78.45
                           72.94
       87.88               42.81
       67.37               31.04
       66.57    233.45     11.55
       57.67     98.43     10.13
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

610 :

                           88.40
                           75.44
                           72.94
                           42.81
                           35.33
                           31.04
       87.88               26.48
       67.37               23.91
       66.57    233.45     23.86
       57.67     98.43     10.13
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

611 :

                           91.36
                           89.47
                           78.45
                           75.44
       87.88               34.60
       67.37               31.04
       66.57    233.45     18.43
       57.67     98.43     11.55
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

612 :

                          234.65
       87.88               95.02
       67.37               58.38
       66.57    233.45     23.86
       57.67     98.43     18.43
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

613 :

                           95.02
                           78.45
                           75.44
                           58.38
       87.88               42.81
       67.37               35.33
       66.57    233.45     26.48
       57.67     98.43     18.43
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

614 :

                           91.36
                           89.47
                           88.40
       87.88               75.44
       67.37               35.33
       66.57    233.45     26.48
       57.67     98.43     23.86
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

615 :

                           91.36
                           88.40
                           75.44
                           58.38
       87.88               35.33
       67.37               31.04
       66.57    233.45     26.48
       57.67     98.43     23.91
     + 48.30   + 49.30   +  7.32
     =======   =======   =======
      327.79    381.18    437.66

616 :

                           89.47
                           75.44
                           72.94
                           58.38
                           42.81
       87.88               34.60
       67.37               23.91
       66.57    233.45     18.43
       57.67     98.43     11.55
     + 48.30   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

617 :

                           91.36
                           89.47
                           72.94
                           42.81
                           35.33
       87.88               34.60
       67.37               31.04
       66.57    233.45     18.43
       57.67     98.43     11.55
     + 48.30   + 49.30   + 10.13
     =======   =======   =======
      327.79    381.18    437.66

618 :

                           89.47
                           88.40
                           78.45
                           35.33
                           34.60
       87.88               31.04
       67.37               26.48
       66.57    233.45     23.91
       57.67     98.43     18.43
     + 48.30   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

619 :

                           89.47
                           88.40
                           75.44
                           72.94
       87.88               31.04
       67.37               26.48
       66.57    233.45     23.91
       57.67     98.43     18.43
     + 48.30   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

620 :

                           95.02
                           89.47
                           78.45
       87.88               72.94
       67.37               35.33
       66.57    233.45     31.04
       57.67     98.43     23.86
     + 48.30   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

621 :

                           91.36
                           89.47
                           78.45
       87.88               58.38
       67.37               42.81
       66.57    233.45     34.60
       57.67     98.43     31.04
     + 48.30   + 49.30   + 11.55
     =======   =======   =======
      327.79    381.18    437.66

622 :

                           89.47
                           88.40
       87.88               75.44
       67.37               72.94
       66.57    233.45     58.38
       57.67     98.43     34.60
     + 48.30   + 49.30   + 18.43
     =======   =======   =======
      327.79    381.18    437.66

623 :

                           91.36
                           89.47
       87.88               75.44
       67.37               72.94
       66.57    233.45     42.81
       57.67     98.43     34.60
     + 48.30   + 49.30   + 31.04
     =======   =======   =======
      327.79    381.18    437.66

StuRat (talk) 20:12, 11 September 2012 (UTC)[reply]

September 10

Matrix subgroup

I'm interested in identifying the subgroup of GL(n,Z) generated by the squares of transvection matrices (i.e. squares of matrices that differ from the identity matrix by replacing a single zero with a one) -- in particular, I'd like to know a presentation. It's not too difficult to see (or maybe it is) that any matrix in this subgroup must have odd entries on the diagonal and even entries elsewhere, and so it is contained in the level 2 principal congruence subgroup (it might perhaps be the whole congruence group?). Any pointers towards a presentation, or just an identification of this group would be appreciated! Thanks, Icthyos (talk) 10:11, 10 September 2012 (UTC)[reply]

Related to your earlier question - Wikipedia:Reference_desk/Archives/Mathematics/2012_March_27#Homomorphisms_from_GL.28n.2CZ.29_into_a_symmetric_group.3F ? The wikilinks there, especially the 4th & 5th section of Special linear group & Steinberg group (K-theory) should help - there's a presentation of SL (n, Z) there. Has to be some modification of the Steinberg relations, and the group you are looking at should (nearly?) generate the congruence subgroup, as the elementary matrices / transvections generate SL. A look at the first chapters of Milnor's book on should help (even if not directly, because it is Milnor).John Z (talk) 09:14, 11 September 2012 (UTC)[reply]

It's not completely relevant to that question, I just always seem to end up mapping into GL(n,Z)! I've realised that I can throw in inversion matrices too if I want, as well as squares of transvections, and I'm *pretty* sure that together they generate the level 2 congruence group (the transvection squares alone, however, do not). So, it comes down to finding a presentation of the congruence subgroup...but surely someone must have done this before? I mean, it's a finite index subgroup, so I can definitely churn out a finite presentation using Reidemeister-Schreier, but...really? Must I? Thanks for the suggestions, Icthyos (talk) 09:51, 11 September 2012 (UTC)[reply]

Standard metrics for quality of multi-objective optimization?

Tried looking through relevant articles on optimization but couldn't find any notice. Is there any "standard" metric to judge the quality (i.e. how close to the Pareto front are the results found) for a heuristic optimization algorithm? This is in the setting of discrete input variables (hence finite state space). A concrete case study is available for direct comparison of the solutions found, but it would be nice to learn if such metrics are (generally) used.

For example, say the Pareto front lies for two variables to be maximized lies on (0,10), (8,8), and (10,5), but the algorithm identifies (0,10), (7,7) and (10,5), thereby missing one solution on the front (and none in between), out of a state space of size 128? Of course such a metric can be "made up" (e.g. share of missed solutions better than the identified Pareto border, area of the space between identified and real border), but I want to know if there is any such metric which is used as a "standard". 130.237.57.87 (talk) 14:53, 10 September 2012 (UTC)[reply]

I don't know if there is a standard for this, but if there is one I would predict that it would take this form: the average, over all points identified by the algorithm, of the squared Euclidean distance to the nearest point on the true Pareto front.

One could simulate the heuristic algorithm, and its resulting average of squared Euclidean distances, over a large number of hypothesized problems for which the entire Pareto front is known, then take an average (of the averaged Euclidean distances) over the set of all hypothesized problems. It seems to me, however, that this approach will be sensitive to the choice of hypothesized problems, so one would have to choose that set carefully based on what sort of problems, with what frequencies, one plans to use the heuristic for.

A possible flaw in the above suggested metric is that it fails to penalize heuristics that don't offer many candidate points Duoduoduo (talk) 17:35, 11 September 2012 (UTC)[reply]

What is a cross-over point in a rational funciton?

My daughter's math class is working with rational functions and graphing them. What is a "cross-over point" of a function such as y = 3x/(x+4)? IS it where the horizontal and vertical asymptotes cross, (-4,3) in this case? Bubba73 ^{You talkin' to me?} 20:53, 10 September 2012 (UTC)[reply]

That sounds reasonable. The only other possibility I can think of is if it refers to the X- and Y-intercepts (both (0,0), in this case). StuRat (talk) 21:12, 10 September 2012 (UTC)[reply]

Her textbook doesn't seem to say and she can't remember what the teacher said. But as I thought about it, it was all I could think of that made sense. They are working on asymptotes. Bubba73 ^{You talkin' to me?} 22:44, 10 September 2012 (UTC)[reply]

While I've never heard that terminology myself, I believe you have given the most reasonable interpretation. One other thought though, it might be a trick question, with the answer being "this curve never crosses over either asymptote" (some curves do). StuRat (talk) 23:53, 10 September 2012 (UTC)[reply]

In a case like this, I just give every bit of info I have:

The 2 asymptotes cross at (-4,3).
The curve does not cross either asymptote.
The X-intercept is at (0,0).
The Y-intercept is at (0,0).

StuRat (talk) 00:01, 11 September 2012 (UTC)[reply]

There were three of these on a test she got back today, so I doubt it was a trick question. (She missed all three.) Bubba73 ^{You talkin' to me?} 00:22, 11 September 2012 (UTC)[reply]

Sorry to hear that. Did any of those curves cross the asymptotes ?

In any case, since this terminology was apparently made up by the teacher, she better ask the teacher what it means, so she will know for the finals. StuRat (talk) 00:24, 11 September 2012 (UTC)[reply]

Do you mean did they cross the X- or Y-axis? Bubba73 ^{You talkin' to me?} 01:43, 11 September 2012 (UTC)[reply]

There were three rational functions. For each they were to graph it, give the x- and y-intercepts, the horizontal and vertical asymptotes, the cross-over point, and the hole. The hole is a point that is not on the graph in the original function, but is after reducing the function by cancelling. She missed cross-over point on all three, and neither of us know what it is. Bubba73 ^{You talkin' to me?} 04:19, 11 September 2012 (UTC)[reply]

No, I meant a curve which crosses it's asymptote(s). Here's an example: [1]. That one crosses many times, but it's also possible to have a curve which crosses it's asymptote just once. StuRat (talk) 05:00, 11 September 2012 (UTC)[reply]

Two more examples of rational functions, whose graphs cross theis asyptotes just once, can be found in Rational function#Examples. --CiaPan (talk) 05:16, 11 September 2012 (UTC)[reply]

Resolved

I got email back from the teacher, and the cross-over point is where the function crosses the horizontal asymptote. Bubba73 ^{You talkin' to me?} 14:42, 11 September 2012 (UTC)[reply]

Interesting that it's just the horizontal asymptote. Why have a special term for this special case, and exclude vertical asymptotes and all others ? This is sounding less and less like a standard mathematical term and more like one invented by the teacher. It's up to you, but you could lodge a complaint that the teacher is using non-standard terms not included in the text, and potentially get those questions excluded from the test. (I was once placed in a remedial math class because I answered all the division problems on a test correctly, but in fractional form, when they wanted them in the much inferior remainder form.) StuRat (talk) 18:05, 11 September 2012 (UTC)[reply]

It was a function like this one File:RationalDegree2byXedi.svg. I don't see the significance of the point where it crosses the horizontal asymptote. The term isn't in the index of the textbook and I read the entire section on rational functions that they had just covered, and it isn't in there. Bubba73 ^{You talkin' to me?} 18:58, 11 September 2012 (UTC)[reply]

Yes, and I didn't find it by Googling either. StuRat (talk) 19:01, 11 September 2012 (UTC)[reply]

(responding to "exclude vertical asymptotes" above) A function can't cross a vertical asymptote. I reread the section and now I see where it comes from. The book gives guidelines for sketching rational functions. One step is to see if it crosses the horizontal asymptote and (if it does) plot that point. But it does not use the "cross-over point" terminology. Bubba73 ^{You talkin' to me?}

Well, a function in the form y = f(x) won't typically cross a vertical asymptote, but one in the form x = f(y) often will, as will those in polar coordinates. Also, there can be asymptotes which are neither horizontal nor vertical. StuRat (talk) 19:51, 11 September 2012 (UTC)[reply]

September 11

A countanle $\pi -weight$

Hi, Does anyone know what is a (countable) A countanle $\pi -weight$ for a topological space? And also, is there a difference between ${\overline {A}}$ to ${\overline {int({\overline {A}})}}$ . In other notation: is there a difference between $Cl(A)$ to $Cl(int(Cl(A)))$ ?. Thanks! Topologia clalit (talk) 11:14, 11 September 2012 (UTC)[reply]

Don't know about pi-weights. For the second question, let A in R be a closed interval union a point. Staecker (talk) 11:35, 11 September 2012 (UTC)[reply]

The π-weight of a space is a minimal cardinality of its π-base.—Emil J. 12:05, 11 September 2012 (UTC)[reply]

R[X]/(X^2-1)

If I have a Field (eg R) and I build the quotient as follows R[X]/(p(x)) with a polynomial p(x), then I get a field extension as usual if p(x) is a minimal polynomial. What happens if p(x) is reducible (ie. not a minimal polynomial)? For example in the cases: R[X]/(X^2-1)

or

R[X]/(X^2)

--helohe (talk) 19:45, 11 September 2012 (UTC)[reply]

The result will not be a field, but a ring with zero divisors. —Kusma (t·c) 20:13, 11 September 2012 (UTC)[reply]