Students arrived in class today to find tic-tac-toe boards on their desks, and a challenge on the board:
A 3×3 square is partitioned into 9 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 90 degrees 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?
This is a scary, scary, looking problem, which I have shared before on the blog. I learn much about my classes by observing the reactions to these sorts of problems: who reads carefully, who dives right in, who turns to share thoughts with their neighbors, who gives up immediately, and so on…so much problem solving comfort revealed in one problem.
So how do we start? After a few reads, I asked students to experiment with their boards, and discover some patterns which meet the problem’s constraints.
As students made discoveries and found boards which met the problem’s requirements, I invited a few up to the board to explain their work. This led more students, many who were apprehensive at the start, to think about the problem and the rotations.
Students began to discuss their findings, and some agreements were reached:
- The center square must be black
- There must be at least 5 black squares
But do ALL grids with at least 5 square work? This led to one last challenge for the day – find a grid with 5 black squares which does not work? This was quickly tackled by a few groups:
We haven’t tackled the randomness and probability aspects of the problem yet – that will resume tomorrow. But hopefully less apprehension over complex-looking problems and some developing teamwork!