Locked Polyomino Tilings

A locked $t$-omino tiling is a tiling of a finite grid or torus by $t$-ominoes such that, if you remove any pair of tiles, the only way to fill in the remaining space with $t$-ominoes is to use the same two tiles in the exact same configuration as before. We exclude degenerate cases where there is only one tiling due to the grid/torus dimensions. Locked $t$-omino tilings arise as obstructions to popular political redistricting algorithms. It is a classic (and straightforward) result that grids do not admit locked 2-omino tilings. In this paper, we construct explicit locked 3-, 4-, and 5-omino tilings of grids of various sizes. While 3-omino tilings are plentiful, we find that 4- and 5-omino tilings are remarkably elusive. Using an exhaustive computational search, we find that, up to symmetries, the $10 \times 10$ grid admits a locked 4-omino tiling, the $20 \times 20$ grid admits a locked 5-omino tiling, and there are no others for any other grid size attempted. Finally, we construct an infinite family of locked $t$-omino tilings on tori with unbounded $t$.