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. The colors are updated as follows: if a white square is in a position formerly occupied by a black square, it is painted black; if a black square is in a position formerly occupied by a white square, it is painted white. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
