Mathematics | Physical Sciences and Mathematics
Dr. Lauren Keough
Given a p×q rectangular board (height p and width q), we may fill in the area with 2×1 tiles. We say the board is tileable if the board can be filled with non-overlapping tiles leaving no open space. Not all boards are tileable, for example a 5 × 5 board is not tileable because it has an odd area. A fault-free tiling exists on the board if every line through the board parallel to a side goes through a tile. Previous work on this subject has been completed by Ron Graham in “Fault-Free Tilings of Rectangles" and Emily Montelius in “Fault-Free Tileability of Rectangles, Cylinders, Tori, and Möbius Strips with Dominoes". Here, we fill in the details of Graham’s proof of exactly which boards have fault-free tilings and give an efficient way to create such tilings.
Dettling, T. Elise and Goetzinger, Amelia, "Fault-Free Tilings" (2022). Undergraduate Research. 6.