Sprague-Grundy values of CRIM, the row/column-deletion game on integer partitions

Abstract

Crim is the impartial game whose positions are integer partitions and whose moves delete one row or one column of the Young diagram, the surviving cells reattaching into a partition again. Bašić, Gottlieb and Krnc determined the Sprague–Grundy values of several families of positions and closed with nine conjectures. We show that one of them is false, and false on the paper's own terms. The conjecture predicts G(Rᵏ_(r,r-1)) for r ≥ 7; on the line k=r-2 the rectair Rʳ⁻²_(r,r-1) is the padded staircase PSᵣ₋₁, an identity recorded in the source itself, so that conjecture and the source's conjecture on padded staircases speak about the same positions and predict different values at every r ≥ 7. Computation decides between them: G(PS₆)=1, G(PS₇)=3 and G(PS₈)=1, where the first conjecture predicts 3, 1 and 3. Its parity condition is inverted. We record four further failures at degenerate parameters, one of them a genuine counterexample to a published theorem: the characterisation of losing three-part partitions fails at [1,1,1], which the statement places in P although the single column may be removed at once. We then verify finite instances of the remaining conjectures, among them the main conjecture (every losing partition has even Dyson rank) for all 18 460 partitions of size at most 28. All statements are machine-checked in Lean 4.

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