Wikipedia:Reference desk/Archives/Mathematics/2023 September 28

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September 28

Probability of a holey cube

Suppose that eight identical cubes, each with a hole bored from the centre of one face to the centre of the opposite one, are assembled with random orientation into a two by two cube. What is the probability of no hole all the way through? Assuming that the smaller cubes are placed and then oriented gives 6561 possible orientations, but I can't find a way of calculating exactly the number without any hole alignment. A counting program gives the fraction 828/6561, reducing to 92/729. Is this correct, and if so is there an easy way to calculate it? Would the method extend to larger combined cubes?2A00:23C6:AA0D:F501:E536:AB5C:96D1:E7CE (talk) 13:55, 28 September 2023 (UTC)

I get the same number of 828 unholey cubes.  --Lambiam 15:27, 28 September 2023 (UTC)

If you view the hole at each corner as a directed edge pointing to another corner, you form a directed graph which is a subgraph of the cube graph. (By this I mean the underlying undirected graph is a subgraph. This has outdegree 1, but variable indegrees. For a general graph, the number of of such graphs is the product of the degrees, in this case 3⁸, and the question is what fraction of them have no 2-cycles. I would apply the Inclusion–exclusion principle; you'd get something like:

$3^{8}-\#{\rm {{edges}\cdot 3^{6}+\#\{{\rm {{pairs\ of\ disjoint\ edges}\}\cdot 3^{4}-\#\{{\rm {{triples\ of\ disjoint\ edges}\}\cdot 3^{2}+\#\{{\rm {{quadruples\ of\ disjoint\ edges}\}\cdot 1}}}}}}}}$

It's then a matter of counting edges (12), pairs, triples and quadruples of disjoint edges. Counting pairs etc. isn't trivial, but I was able to do it without resorting to a counting program. I get 42 pairs of disjoint edges, 44 triples, and 9 quadruples to give a final value of 3⁸-12⋅3⁶+42⋅3⁴-44⋅3²+9=828.

I don't know how easily this approach generalizes to other collections of cubes; I saw it as a problem in graph theory. There may be easier ways of doing it too; this is just the first one I through of. --RDBury (talk) 15:56, 28 September 2023 (UTC)

(edit conflict) I got along a similar road (but got stuck at the step parallel to "counting pairs etc."). Here’s my work anyway for a non-graph formulation:

Let $P_{n}$ be the probability that there are n through-holes for a uniform random selection of the configuration of eight minicubes. We are interested in $P_{0}$, but it seems easier to count the high-through-hole-count configurations. Because $P_{n\geq 5}=0$, there are five nonzero $P_{n}$; let’s find out some relationships between them.

• By the definition of probability, $\sum _{n}P_{n}=1$.
• $\sum _{n}nP_{n}=4/3$. Proof: the left-hand-side value is the average number of through-holes in a given configuration. By the linearity of expectation, that is equal to the probability that a given pair of adjacent cubes are co-aligned, times the number of such pairs. Each pair of adjacent cubes is co-aligned with probability 1/9 (each cube has probability 1/3 of having its hole pointing to the other), and there are 12 such pairs (4 along each dimension). 12*1/9 = 4/3, QED.

Unfortunately, I do not think there is any useful interpretation of the higher-order moments that would allow us to calculate one of them. Therefore, we still have to find out $P_{2},P_{3},P_{4}$ to compute $P_{0}$ out of the two equations above.

The four-hole configurations are achieved by having all holes aligned in a single direction. There are three possible directions (denominator); the numerator is just the number of configurations. Hence $P_{4}={\frac {3}{3^{4}}}$.

I could not calculate $P_{3}$ with my spatial visualization skills, let alone $P_{2}$. But maybe someone can do it or find another equation linking the $P_{n}$ together. Tigraan^{Click here for my talk page ("private" contact)} 15:59, 28 September 2023 (UTC)

This looks like and interesting approach. Perhaps if might be worthwhile to compute the probabilities and moments from what is known so far and work backwards to find interpretations. Fwiw, a quick search of 1, 12, 42, 44, 9 in the OEIS turned up A302235. (There were other hits but this one was relevant.) --RDBury (talk) 17:24, 28 September 2023 (UTC)

Help with finding the angle

Hello, I need help with this problem:

Given a right triangle ABC where θ is one of its acute angles, and Δx and Δy represent small changes in the x and y coordinates of point C as it moves along the hypotenuse, determine an expression for the tangent of angle θ in terms of Δx and Δy.

How should I start off this problem?

TheAlienMan2002 (talk) 19:12, 28 September 2023 (UTC)

I'm pretty sure there isn't enough information to solve the problem. You could find the angle of the hypotenuse to the x-axis, but that doesn't tell you how the sides are oriented. Is there an assumption you haven't included, such as that the sides of the triangle lie on the axes? Even with that, "one of its acute angles" isn't specific enough; there are two complimentary acute angles and we don't know which one is θ. --RDBury (talk) 20:26, 28 September 2023 (UTC)

Usually, when one writes "a right triangle ABC", the right angle is at the vertex labelled C, which is not on the hypotenuse. If one vertex, being one of the end points of the hypotenuse, moves along the hypotenuse while the other two are stationary, the triangle ceases to be a right triangle. Is that the intention?

If you scale up the triangle, keeping the angles the same, the same Δx and Δy changes will give a smaller change in θ, so the problem is underdetermined.  --Lambiam 21:18, 28 September 2023 (UTC)

Opps, wrote the wrong math equation. This is the correct problem, the other one was a visual.

In a right triangle XYZ, θ is one of its acute angles. As point Z moves along the hypotenuse, the x-coordinate (Δx) and the y-coordinate (Δy) of point Z change. The length of the hypotenuse is 10 units, and Δx and Δy are given as follows:

• When Z moves from its initial position to a new position, Δx = 3 units.
• During the same movement, Δy = 4 units.

Determine the tangent of angle θ in terms of Δx and Δy for this right triangle. TheAlienMan2002 (talk) 21:43, 28 September 2023 (UTC)

The vertex of a triangle cannot "move along the hypotenuse". This is a very poorly written question whose meaning is not obvious. You should go back and ask whoever posed the question to clarify it. –jacobolus (t) 22:03, 28 September 2023 (UTC)

A point can move along the hypotenuse. TheAlienMan2002 (talk) 22:08, 28 September 2023 (UTC)

You can't have the vertex of the triangle move along the triangle's own side. That's a nonsensical statement. You could label some different moving point along the hypotenuse. –jacobolus (t) 23:43, 28 September 2023 (UTC)

Well yeah, the label is point Z and it's moving along the hypotenuse. TheAlienMan2002 (talk) 00:01, 29 September 2023 (UTC)

It doesn't matter if Z is moving or not, this version of the problem has the same issue as the previous version, so not enough information. In general there shouldn't be any information in a picture or diagram that's not written out in words. Diagrams are to help understand the description, but they can't replace it. Also, per the guidelines at the top of the page, this seems to be homework so we can only get you started with a solution. --RDBury (talk) 03:56, 29 September 2023 (UTC)

Let $X$ be the vertex of the right angle, so the hypotenuse is the line segment $YZ,$ while $\angle YXZ={\tfrac {\pi }{2}},$ $\angle XZY=\theta$ and $\angle XYZ={\tfrac {\pi }{2}}-\theta .$ Also, let $Z'$ represent the "moved" vertex, a point on the line segment $YZ,$ so right triangle $XYZ$ is transformed into a non-right triangle $XYZ'.$ We have a new angle $\theta '=\angle XZ'Y.$ If the position of $Z'$ differs from that of $Z'$ by a vector $(\Delta x,\Delta y)=(3,4),$ the length of the line segment $Z'Z$ equals $5,$ half the length of $YZ,$ so $Z'$ is the midpoint of the hypotenuse. Then $XYZ'$ is an isosceles triangle, and $\angle YXZ'=\angle XYZ'=\angle XYZ={\tfrac {\pi }{2}}-\theta ,$ so $\theta '=\angle XZ'Y=\pi -(\angle YXZ'+\angle XYZ')=2\theta .$ If $\tan \theta =t,$ $\tan \theta '={\frac {2t}{1-t^{2}}}.$

A setting that is consistent with the given data is that the vertices of the original triangle has Cartesian coordinates $X=(6,0),$ $Y=(6,8),$ $Z=(0,0),$ $Z'=(3,4).$ Then $\tan \theta ={\tfrac {4}{3}}$ while $\tan 2\theta =-{\tfrac {24}{7}}.$ There are different settings that are also consistent with the given data.  --Lambiam 15:15, 29 September 2023 (UTC)