# Wikipedia:Reference desk/Archives/Mathematics/2007 August 20

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# August 20

## Null day (not)

Is there nothing notable that that could be added to make this date notable? - hydnjo talk

Today is notable. A few selected anniversaries can be found at Wikipedia:Selected anniversaries/August 18. --Iamunknown 21:37, 19 August 2007 (UTC) Thank you!

August the 18th of year 2007

Can be rewritten as 18082007

Which is a prime number. Very Very Notable. 202.168.50.40 01:04, 20 August 2007 (UTC) Thank you!

08182007 is also prime. There are 13 days DDMM for which DDMM2007 and MMDD2007 are distinct primes: 0301, 1802, 0103, 2203, 1904, 0806, 2606, 1207, 3107, 0608, 1808, 2609, 0712. PrimeHunter 01:22, 20 August 2007 (UTC) Thank you!

Not strictly mathematical but what the hey: Messier object M18 has an apparent magnitude of 8.0 (Thus 8/18, which is kind of a stretch.) WTF? - hydnjo talk

Wow - this indeed has turned out to be an interesting day, can't hardly wait for the next null day! - hydnjo talk

## 36 an achilles number

I believe 36 is an achilles number. it is:

${\displaystyle 2^{2}\times 3^{2}}$

so it is powerful, and it is

${\displaystyle 6^{2}=m^{k},m=6,k=2}$.

However, the wikipedia article does not list it in the first 5000. Also, the article does not mention 64, which also meets criteria.

Am I missing something? Should these and/or others be added?Micah J. Manary 03:20, 20 August 2007 (UTC)

Reread the article... it says powerful numbers that are not perfect squares are Achilles numbers. Gscshoyru 03:29, 20 August 2007 (UTC)

Yikes! thanks Micah J. Manary 03:40, 20 August 2007 (UTC)

## Prime[n] = 2 n + 1

Is there a nth prime number which is also 2*n+1

202.168.50.40 03:47, 20 August 2007 (UTC)
Yes, there is at least one less than 100. But see prime number theorem. --KSmrqT 04:00, 20 August 2007 (UTC)
My guess is there are exactly two such numbers existing.
That's correct. Now go and prove it; it's good for the soul. Algebraist 15:31, 21 August 2007 (UTC)

## Math questions re child’s game “boxes” or “houses”

Don’t know what US people call this game. We played it as kids and it so simple anyone can join in. You make a grid of pencil lines say about 5 across and down, so you get 25 boxes. (You can start with bigger or smaller grids). There are two players. First player fills in just one interval - any one they like – and this is like building the side of one box. Second player does likewise, and they continue to take turns. If Player A fills in an interval which is the 3rd side of a box, the Player B can fill in the interval which represents the 4th and last side of the box, thus completing it, and thereby owning it. Player B then writes his initials in the box, and fills in another interval, and it is A's turn again. The object of the game is to obtain as many boxes as you can. When the game is finished the players and the grid is completely full of boxes with players’ initials on them, they count how many boxes each has, and the winner is the player with the most.

During what could be termed the “free” section of the game, the players need do little more than fill in intervals anywhere on the grid, taking care only not to fill in the 3rd side of any box. After a time, there will be no free moves left, and the players will be forced into filling in 3rd sides of boxes. Then the strategy becomes one in which each player tries to give the other player the absolute minimum of boxes. I am not a mathematician, not even an amateur one, but thinking about this recently, I came to the conclusion that there was a lot more to this simple child’s game than might first meet the eye. My questions are as follows:

1. Because they are familiar to everyone, let’s take a Sudoku – style grid of 9 X 9, i.e. a grid of 81 squares. Suppose a computer program fills in intervals in this grid randomly, working only on the rule that it cannot fill in any interval such that it would represent filling in the 3rd side of a box. When there are no more moves possible, the program stops. Is the number of intervals filled always the same? Or does it depend on the configuration of filled intervals? If the latter, then what are the minimum and maximum number of intervals than can be filled before the process of filling in has to end?

2. Suppose now, that we have TWO computer programs which are to play each other by the rules described above. Program Alice is to play randomly in the initial “Free” stage of the game, while Program Bob is permitted to use any strategy the programmer likes to maximise the number of boxes Bob will have at the end of the game. Can Bob obtain an advantage over Alice using logical strategy? And if so, what is the best strategy to maximise the number of boxes won?

3. Now imagine that both Programs Bob and Alice are smart and both work on “best possible move” strategies. What is the outcome now? Does it depend only on who moves first, no matter how large the initial grid? If both make “best possible move”, then does the configuration of the game when there are no more free moves left always look the same. As the game cannot be a draw, then how large would the margin be between the winner and loser? Does this, the simplest of children’s games, now reveal itself as more akin to the Chinese game Go, the last game to resist computerisation, and regarded as one of the most difficult games ever created? Myles325a 04:41, 20 August 2007 (UTC)

The version I know of is called Dots and Boxes. There has been a lot of mathematical analysis of it. See the following links: [1] [2] [3]. —David Eppstein 04:50, 20 August 2007 (UTC)
Just addressing the very first question (part of #1): the length of the "free" phase certainly can vary. Consider a 3×3 grid of boxes (4×4 of points): drawing every vertical line gives us 12 segments, but drawing the entire periphery and then two parallel segments in the middle gives us 14. --Tardis 21:05, 20 August 2007 (UTC)
Quick answers to the other questions (though there is a lot to be said about this game, so you should check out the websites above and/or find a book): 2. Yes, a perfect computer program (or even an awake human) can massacre random play (unless you randomly get a perfect strategy, of course). The best strategy will be rather complicated.
3. In general, the result of the game with perfect play is unknown and depends on the size of board. I don't know if 9x9 is solved yet (but the book I read is 25 years old!). If it is, the best strategy will be very complicated. For the 2x2 game (with the second player winning draws), for example, the first player can force a 3-1 win but it's quite complicated and (against good defense) requires a sacrifice: i.e. your idea of the 'free period' is flawed, as sometimes the best move is to give away points in return for greater gains. Algebraist 21:34, 20 August 2007 (UTC)
Let me try to finish #1, since Algebraist covered the other two. Since each of mn squares must have no more 2 of its borders drawn, and a segment can provide a border to just 1 (if on the outside) or 2 squares, we see that there must be ${\displaystyle s\leq 2mn}$ segments drawn; moreover, since the periphery has only ${\displaystyle 2(m+n)}$ segments to it, we can reduce the upper bound (for ${\displaystyle mn>1}$, where the whole periphery may be used) to ${\displaystyle mn+m+n}$. (This isn't tight: the 3×3 game cannot have 15 segments drawn, just 14 as I gave earlier.) It is tempting to propose a lower bound of ${\displaystyle s\geq mn}$, as a segment cannot provide more than 2 box-edges and there are ${\displaystyle 2mn}$ to provide, but it is possible to go lower: a box may have no edges drawn because any one of them would give an adjacent box 3. For example, drawing the inner two edges of each corner box in a 3×3 game is just 8 edges. It would be possible to get some sort of lower bound based on the maximum frequency of such deficient boxes, but it would probably still not be tight. --Tardis 15:31, 21 August 2007 (UTC)

Well, I have now played about 40 dot and box games against the computer [4] on the lowest level and smallest grid, and been beaten every time. But I gather the strategy is one of leaving yourself boltholes you can use to sacrifice a point, and thus giving the lead back to your opponent when this is advantageous for you. It seems like a mirroring strategy. As one player designs small areas he can use to make small sacrifices, the other player compensates by doing the same thing. A big part of the strategy seems to be not to caught “breaking the symmetry” at the wrong time. I was thinking what a 3 dimensional version of this would look like. Now, that would certainly be testing the limits of computability. The object of the game would be to win the larger number of cubes (not boxes) and as these have 12 sides as opposed to 4 for a box, the strategies would be simply mind-boggling! (Actually, just tried to find something on 3D dots and box, but nothing in google. Anyone got ideas? Myles325a 01:07, 22 August 2007 (UTC)

## Wikipedia:Reference_desk/Miscellaneous#6_shapes_puzzle

Does anyone recognise the puzzle describe above - the description isn't very clear..

Must be some sort of maths puzzle??87.102.2.76 12:07, 20 August 2007 (UTC)

If it's the one at [5], it appears to be six puzzle pieces that form a cube. There are 6!*6^4/(6*4)=5!*5^4=75000 possible arrangements. I just felt like analyzing it. — Daniel 22:13, 21 August 2007 (UTC)

## Constructing polytope from Schläfli symbols

I'm trying to understand how Schläfli symbols work, especially how one constructs a polytope given its symbol. Could anyone point me towards an algorithm for generating the coordinates of the vertices, edges, etc.? I'm only really interested in doing this in 4d, but a general algorithm would be nice. (My maths knowledge mostly consists of linear algebra and calculus, though I'm starting an informal course on topology soon, so keep it simple). --Taejo|대조 14:02, 20 August 2007 (UTC)

## Simplification of integral

Consider the integral below:

${\displaystyle \int _{0}^{1}\!A\left(s\right){e^{\int _{0}^{s}\!A\left({\it {s_{1}}}\right){d{\it {s_{1}}}}}}{ds}}$

Is there a smart way, maybe using integral by parts, to simplify this integral so that the nested integral is removed? A is a matrix, but that shouldn't make much difference. Regards, deeptrivia (talk) 20:19, 20 August 2007 (UTC)

The derivative of ${\displaystyle F(s)=\int _{0}^{s}\!A\left({\it {s_{1}}}\right){d{\it {s_{1}}}}}$ is ${\displaystyle F'(s)=A(s)}$ so what you have is ${\displaystyle \int _{0}^{1}F'(s)e^{F(s)}ds=e^{F(1)}-e^{F(0)}=e^{\int _{0}^{1}\!A\left({\it {s_{1}}}\right){d{\it {s_{1}}}}}-1}$. Stefán 01:52, 21 August 2007 (UTC)
Oh, I guess the answer is simply ${\displaystyle e^{\int _{0}^{1}\!A\left({\it {s_{1}}}\right){d{\it {s_{1}}}}}}$ Is that correct? deeptrivia (talk) 01:52, 21 August 2007 (UTC)
Heh, funny, after five hours there are two answers at the same minute. But you missed a -1, right? Stefán 01:54, 21 August 2007 (UTC)
Sure, it is. It was a silly question to begin with, now that I think of it. deeptrivia (talk) 01:56, 21 August 2007 (UTC)
Thanks, you beat me! deeptrivia (talk) 01:53, 21 August 2007 (UTC)
One question. A is a matrix. Can I just replace the " - 1" in the final answer with " - I", the identity matrix? deeptrivia (talk) 01:53, 21 August 2007 (UTC)
Yes, ${\displaystyle F(0)}$ is then the zero matrix and exp of that is the identity. Stefán 01:55, 21 August 2007 (UTC)
Yup, thanks a lot! deeptrivia (talk) 01:58, 21 August 2007 (UTC)

## What area of mathematics do I study now?

In school I've followed the (American) progression of algebra, geometry, algebra II (polynomials, exponents/logarithms, and a cursory examination of complex numbers and probability), trigonometry and calculus (differential and integral; largely single-variable, also with vectors, more probability, some statistics, and sequences/series). What's the next step? Linear algebra? Topology? Is there a definite next step, or have I pretty much covered the basics and do I have choice now in what area to pursue? I guess what I'm asking is: after single-variable calculus, what are the next courses that a college student would take? Any help is appreciated. Strad 23:54, 20 August 2007 (UTC)

The typical series of instruction (assuming a science/engineering education as opposed to a business or liberal arts education) will include three semesters of Calculus at the college level. If you took one year of AP Calculus in High School (Calc AB), that usually translates to the first semester. A second year (Calc BC) gives you the second semester (which sounds like roughly where you are). The third semester is basic multi-variable calculus. From there, it continues to Linear Algebra and Differential Equations. Some schools have an enforced order of Linear Algebra followed by DEs, others offer them as independent courses and less frequently, there's a two semester sequence which combines the two. From there, you would often take a semester of Analysis, either Math Analysis I (it's usually a two-semester sequence) or Fourier Analysis (if you're more applications-minded). But a lot depends on your major/interests. In the social sciences, the math would be skewed towards stats. Business majors take "business calculus" which often has little of either business or calculus (it tends to focus on integrals and derivatives of polynomials and in the textbooks that I've seen, proposes some absurd models in the name of mathematical pedagogy. Many schools offer a class called Finite Math which is an interesting mix of basic linear algebra and combinatorics among other things which is much more useful than Business Calculus. I used to teach a class called "Math for Liberal Arts Majors" which had some stats, some probability, some consumer/business math and some geometry (and a lousy textbook).
Taking a look at the catalogs of colleges that you're contemplating attending will give you a clearer picture of your mathematical future. Donald Hosek 00:06, 21 August 2007 (UTC)
(edit conflict) Although I can't speak as to the specifics of the US education system, in general there are several directions you could pursue. I would suggest that linear algebra is a good idea, as is something involving differential equations. Topology is a subject with a surprising amount of depth and complexity which, depending on what kind of things the course covers, is really only something that should be considered once you have a bit more grounding in algebra (by which I'm referring to algebraic constructs like fields, rings and groups, which will probably follow on from linear algebra) and possibly some differential geometry. You could also go off on a completely different tangent and look at something like discrete mathematics or logic. Confusing Manifestation 00:08, 21 August 2007 (UTC)
Thanks to both of you. Strad 01:07, 21 August 2007 (UTC)
The algebra of fields, rings, and groups would be necessary for algebraic topology, but elementary point-set topology (which is almost always covered first) doesn't use them at all. Tesseran 05:05, 21 August 2007 (UTC)
But IMO (and that of my university) it would be crazy to study point-set topology without some grounding in elementary real analysis so the topology comes across as clever generalisation rather than baffling abstraction. (aside: in the US does basic analysis, with epsilons and deltas and the IVT and the MVT and so on, come under Calculus or Math Analysis I or what?) Algebraist 15:26, 21 August 2007 (UTC)
That basic analysis is (the more advanced) part of Calculus I, a freshman-level college course. Limits are introduced first, typically algorithmically, then given their rigor with e-d, then continuity and such are defined in terms of limits and bring the IVT. (Such properties as surjectivity are covered with continuity or sometimes at the very beginning.) Derivatives are then defined (also in terms of limits, of course) and then bring the MVT with them. --Tardis 16:52, 21 August 2007 (UTC)
Tesseran is right, but that kind of abstract algebra is very useful for general enrichment before studying topology. I recently did a unit in point set topology, and being the only one who had covered groups or set theory, I was at a distinct advantage. I would recommend some general reading in mathematical logic, and doing at least the early chapters of a group theory textbook (and solving the end-of-chapter problems) and also a bit of set theory before tackling topology. Otherwise, it might seem like chasing ghosts (it's very abstract). After that intro stuff, it would be the best thing for someone who has a pure interest in maths, rather than an inclination towards any career. For set theory, I used the Schaum's outline, and have found this to be the most practical and best for a beginner. Likewise, the Schaum's outline for Topology is excellent (I think by the same author). 203.221.126.124 15:28, 21 August 2007 (UTC)
I'm a bit biased, being an algebraist and number theorist at heart, but I think that Abstract Algebra is the best starting point for upper-division math. To answer Algebraist's question, in theory, epsilon-delta, IVT and MVT are taught in Calculus, but since many US students get their Calc I in HS (and sometimes Calc II), they seem to either never learn this well or forget it, so it gets revisited in Math Analysis. The distinction between the two is not always clear and Michael Spivak's Calculus is as often used as an introductory math analysis text as a Calculus text. And I would agree that some grounding in analysis is helpful for understanding topology, although the reverse can also be said: Topology can be helpful in understanding analysis, especially Complex Analysis. I like the Munkres book for learning Topology, although it is helpful to have access to a topologist if one is engaged in self study. Donald Hosek 16:55, 21 August 2007 (UTC)
Oh, certainly. At Cambridge we have a course of real analysis, then a short introduction to point-set topology, then more real analysis, complex analysis and so on. In answer to the original question, I agree with Donald Hosek that basic group theory is probably the friendliest route into serious abstract mathematics, if that's the way you want to go. Algebraist 17:50, 21 August 2007 (UTC)

If you liked calculus, you could start studying analysis seriously. Some topology is useful, but point-set topology in the metric case is more than enough for a lot of analysis. Essentially, it's enough to know what open and closed means, and that you can partition unity in normal spaces. Spivak's Calculus on Manifolds or Munkres' Analysis on Manifolds will teach you multivariate calculus rigorously, and they cover the necessary topological background. From manifolds in euclidean spaces, you will get motivation algebraic topology, but you will need to study some basic abstract algebra before. If you want to keep avoiding algebra, you can escape into differential and functional analysis, although you will need to know linear algebra. Complex analysis is also a possibility. I know many dislike it, but Conway's Functions of One Complex Variable will take you from basic treatment of complex numbers and metric space topology all the way to Riemann surfaces and sheaves; that is quite a tour of higher math. Also, the book is notoriously slow-paced (which is one reason some can't stand it). Phils 21:28, 21 August 2007 (UTC)