A removal puzzle is dealt by taking a finished board apart, and the order that does it is the solution

Fact

Where every move removes pieces — mahjong solitaire, pair-matching boards, peg-style clears — do not place pieces and then test whether the board can be cleared. Run the game backwards:

1. Fill every slot with an unknown piece.
2. Find the slots that are legal to take right now, pick as many as a move consumes, and assign them a matching set.
3. Remove them and repeat until the board is empty.

Replaying those removals forward is a legal game, so the arrangement they describe is solvable by construction. The generation order is not a by-product: it is a certificate, worth returning with the deal and replaying in the test suite move by move.

The one failure mode is the sampler stranding itself — a state with pieces left but fewer legal slots than a move needs, such as two tiles where one sits on the other. It is not a bug in the layout, so budget attempts and redraw with a fresh stream rather than trying to prove it cannot happen.

Why it matters

Generate-then-verify needs a solver, and a solver over a board with tens of thousands of orderings is the expensive part of the program. Unbuilding needs only the legality test the game already has, runs in milliseconds, and cannot return an unsolvable board at all.

How to apply

• Assign pieces at removal time, never beforehand; that is what keeps every step legal.
• Keep the certificate and assert it: replay it through the real move validator and require the board to reach empty.
• Reuse the same routine for an in-game reshuffle, over the slots and pieces still standing, so the rest of the game stays solvable too.
• Solvable is not the same as worth playing — see [[solvable-is-not-the-same-as-worth-solving]] — so also measure how often a player taking a random legal move clears it. Near 100% means the board is not a puzzle. [[no-guess-puzzle-generation]] is the counterpart for puzzles where the question is deduction rather than ordering.