A locked $t$-omino tiling is a grid tiling 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 overall due to small dimensions. Locked $t$-omino tilings arise as obstructions to widely used political redistricting algorithms in a grid model of redistricting. It is a classic (and straightforward) result that finite 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, 4- and 5-omino tilings are remarkably elusive. Using an exhaustive computational search, we completely enumerate all locked tilings on grid sizes up to $20 \times 20$, and all symmetric locked tilings on grid sizes up to $35 \times 35$. We find only a single 4-omino tiling (on the $10 \times 10$ grid) and a small handful of 5-omino tilings (only on $20 \times 20$ grids and larger). Finally, we construct a family of infinite periodic locked $t$-omino tilings with unbounded $t$ for both square and triangular grid lattices.

翻译：锁定$t$联骨牌平铺是一种由$t$联骨牌构成的网格平铺，其特性为：若移除任意两块骨牌，则填补剩余空间的唯一方式必须以完全相同的构型重新使用这两块骨牌。我们排除因尺寸过小导致整体仅存在单一平铺的退化情况。在基于网格的选区重划模型中，锁定$t$联骨牌平铺可作为广泛使用的政治选区重划算法的障碍。经典且直接的结论表明，有限网格不存在锁定2联骨牌平铺。本文构造了不同尺寸网格上的显式锁定3联、4联及5联骨牌平铺。尽管3联骨牌平铺数量丰富，但4联与5联骨牌平铺极为罕见。通过穷举计算搜索，我们完整枚举了尺寸不超过$20 \times 20$网格上的所有锁定平铺，以及尺寸不超过$35 \times 35$网格上的所有对称锁定平铺。结果仅发现单个4联骨牌平铺（位于$10 \times 10$网格）及少量5联骨牌平铺（仅出现在$20 \times 20$及以上网格）。最后，我们在正方形网格和三角形网格格点上构造了具有无界$t$值的无穷周期锁定$t$联骨牌平铺族。