Question

In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is 1/n, where n is a positive integer. Find n.

Picture coming. @TuringTest @mathmale

Answer 1

So far I've gotten that the middle square must be blue, and each spoke of the wheel must have 1 blue square and 2 red squares.

Answer 2

there are 13 choose 8 = 13 choose 5 ways to distribute the remaining red or blue squares, respectively. find the number of ways the pattern you described can be formed, and their ratio should be the probability.
That's my reasoning, let's see if mathmale agrees

Answer 3

Wow! Challenging question!

I'd like to suggest that we build upon your assumption, i. e., that we assume that the central square be blue.

One way of approaching this problem would be to draw all of the possible coloring schemes (which I think would be easier and more concrete than trying to do this problem theoretically). Once you'd done that (assuming that you were willing to do so), you could then turn each one 90 degrees clockwise and determine whether or not it looks the same as before you turned it.

Now supposing that you found that 2 coloring schemes out of a total of 12 satisfy that critieria. then the probability that the scheme you draw at random would look the same when rotated 90 degrees clockwise would be 2/12, or 1/6.

I'm sure you realize that my " 2 " and " 12 " are strictly examples, and are not meant to represent the actual problem that you're dealing with.

Answer 4

I'm reasoning that the blue could be in any square of the three, but it must be in the corresponding square in all 4 spokes. How does this play into the probability?