Problem:
In the array of squares shown below, squares are colored red, and the remaining 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 around the central square is , where is a positive integer. Find .
Solution:
The colors of the cells in the array can be chosen in ways. For an array to satisfy the given symmetry condition, the center square must be blue, and each of the three-square "L" shapes must contain red squares and blue square. Once one of the L's is colored, the other three L's must each be a rotation of the first L. Because red squares and blue square can be placed in one of the L shapes in ways, there are only such arrangements. Therefore the probability that an array satisfies the given symmetry condition is . The requested denominator is .
The problems on this page are the property of the MAA's American Mathematics Competitions