Take an n×n×n cube made of unit cubes. Remove one unit cube at a time, but only if it has exactly d exposed faces — a specific number of neighbors. Can you dismantle the entire cube this way?
Barát and Wanless (arXiv: 2603.23913) connect this combinatorial puzzle to two deeper structures. Going forward, the dismantling process defines a sequence of valid removals. Going backward — adding cubes instead of removing them, with each new cube having exactly d neighbors — the process is bootstrap percolation, the model for how crystals grow and infections spread.
The “perfect solutions” — dismantling sequences where every move is balanced (removes equal amounts from each layer) — correspond to Latin hypercubes. But almost all Latin hypercubes don't correspond to valid solutions. The combinatorial constraint of the dismantling process is strictly stronger than the combinatorial constraint of the Latin structure. Having the right counting property isn't enough; the ordering of removals must also be feasible.
The through-claim: the reverse of destruction is growth, and both are constrained by the same geometry. The dismantling and the percolation reach the same maximal positions — the set of cubes that can be reached by either process is identical, regardless of the order of operations. The geometry of the cube determines which configurations are accessible; the dynamics merely determine the path. Latin hypercubes provide the right combinatorial skeleton, but feasibility requires more — it requires that the skeleton support a valid temporal sequence.
Barát & Wanless, 2603.23913. Combinatorics / bootstrap percolation / Latin squares / hypercubes.