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 $t$.

翻译：锁定t-联骨牌铺砌是指有限网格或环面上的一种t-联骨牌铺砌，其中若移除任意一对骨牌，则用t-联骨牌填充剩余空间的唯一方式是使用相同两块骨牌且保持完全相同的构型。我们排除了因网格/环面尺寸仅存在单一铺砌的退化情况。锁定t-联骨牌铺砌作为阻碍流行政治选区重划算法的反例出现。经典（且直接）结论是网格不存在锁定2-联骨牌铺砌。本文构造了不同尺寸网格中明确的锁定3-、4-和5-联骨牌铺砌。尽管3-联骨牌铺砌数量丰富，但4-和5-联骨牌铺砌却异常罕见。通过穷举计算搜索发现，在考虑对称性的情况下，$10 \times 10$网格存在锁定4-联骨牌铺砌，$20 \times 20$网格存在锁定5-联骨牌铺砌，而其他所有尝试的网格尺寸均不存在此类铺砌。最后，我们在环面上构造了一个具有无界t的无限族锁定t-联骨牌铺砌。