Detecting randomness by counting tiles (2011)

Solution to the bathroom tile problem
Published in the Princeton MAE Departmental Bulletin, 10/2011

The bathroom tile problem asks: when a tile floor (2D truncated plane) is composed of a grid of tiles in which each location is occupied by a tile of random color what is the longest span between two of the same colored tiles in either dimension along the floor. This work addresses how this longest span scales with the relative concentrations of the two tile species. Herein we provide both an analytical expression and Monte Carlo type simulation of bathroom tile floors. The punchline: the bathroom floors in the men's rooms of the engineering quad at Princeton, while lacking a repeating unit are, to a great degree of confidence, non-random.