2012 AMC 10A Problems/Problem 20

The following problem is from both the 2012 AMC 12A #15 and 2012 AMC 10A #20, so both problems redirect to this page.

Problem

A x 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 the grid is now entirely black?

Solution

First, the middle square is always black because it is rotated onto itself. Then we can consider the corners and edges separately. Let's first just consider the number of ways we can color the corners. There is case with all black squares. There are four cases with one white square and all work. There are six cases with two white squares, but only the with the white squares opposite from each other work. There are no cases with three white squares or four white squares. Then the total number of ways to color the corners is . In essence, the edges work the same way, so there are also ways to color them. The number of ways to fit the conditions over the number of ways to color the squares is