The switch operators and push-the-button games: a sequential compound over rulesets

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Problem: Reconstructed statement: In normal-play Push-Cram on an empty $3\times(2k+1)$ grid, players first may place only vertical dominoes; at any turn, including the first, a player may instead push the button, which counts as a move and permanently changes the rules so that only horizontal dominoes may be placed. The conjecture asserts that for every $k\ge0$ this starting position is an $\mathcal N$-position and is a bluff game, meaning every legal first move is winning.

Result: The conjecture is false.

Take $k=1$, i.e. the $3\times3$ grid. A legal first move is to push the button immediately. The remaining game is then horizontal Cram on three independent rows of length $3$.

A single $1\times3$ row has Sprague-Grundy value $1$: any horizontal domino move leaves only one isolated cell, hence no further move. Therefore the position after the immediate push has value

so it is an $\mathcal N$-position for the next player, namely the second player. Explicitly, after the first player pushes, the second player plays in one row; whatever row the first player next plays in, the second player plays in the remaining untouched row and wins.

Thus pushing the button is a losing first move on the $3\times3$ board. Since pushing is legal as a first move under the paper’s definition, the $3\times3$ board is not a bluff game.

This is not merely the endpoint $k=0$. Also, if one tried to repair “first move” to mean “first vertical-domino move,” that repair still fails on $3\times5$: after any one vertical domino, the second player can push. The two affected rows have identical split contributions, which cancel by XOR, while the untouched row has value $\mathcal G(5)=0$ for $1$-row Cram. Hence the pushed position has value $0$, so that vertical first move is losing.

Audit: $k=1$ satisfies $k\ge0$; the button is explicitly a legal move before any domino is placed; the game is finite and impartial, so Sprague-Grundy theory applies. The conclusion directly contradicts the claimed bluff property, hence the stated conjecture is false.

Citation: Definitions and conjecture: Duchêne, Heinrich, Larsson, Parreau, “The switch operators and push-the-button games,” arXiv:1707.07966. Counterexample above is self-contained.