Photo taken in my parking garage this morning. Lots of cutesy titles suggested themselves, among them:
I am a college professor at Milwaukee School of Engineering, where I teach literature, film studies, political science, and communication. I also volunteer with a Milwaukee homeless sanctuary, Repairers of the Breach, as chair of the Communications and Fund Development Committee.
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.
First, look for invariants. The center, unaffected by rotation, must be black. So automatically, the chance is less than Note that a rotation requires that black squares be across from each other across a vertical or horizontal axis.
First, there is only one way for the middle square to be black because it is not affected by the rotation. 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.
We proceed by casework. Note that the middle square must be black because when rotated 90 degrees, it must keep its position. Now we have to deal with the following cases: Case 1: 0 white squares. There is exactly way to color the grid this way. Case 2: 1 white square.