Problem:
A square is partitioned into unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
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Solution:
There are possible initial colorings for the four corner squares. If their initial coloring is , one of the four cyclic permutations of , or one of the two cyclic permutations of , then all four corner squares are black at the end. If the initial coloring is , one of the four cyclic permutations of , or one of the four cyclic permutations of , then at least one corner square is white at the end. Hence all four corner squares are black at the end with probability . Similarly, all four edge squares are black at the end with probability . The center square is black at the end if and only if it was initially black, so it is black at the end with probability . The probability that all nine squares are black at the end is .
The problems on this page are the property of the MAA's American Mathematics Competitions