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arXiv · 2606.16828

CRIM: A Natural Game on Integer Partitions

Ina Bašić·Eric Gottlieb·Matjaž Krnc
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Abstract

We analyze Column-Row Impartial Merge (CRIM), an impartial combinatorial game played on integer partitions. A move in CRIM consists of removing an arbitrary row or column from the corresponding Young diagram, with the remaining parts reattaching to form a single partition. We define rectairs -- a common generalization of rectangles and staircases -- and characterize their $\mathcal{P}/\mathcal{N}$-status. We define the meld operation on partitions and show that the meld of losing rectairs is losing. We introduce Odds-Are-Even (OAE) and Evens-Are-Odd (EAO) partitions, proving that all OAE partitions are $\mathcal{P}$-positions and characterizing the losing positions within EAO partitions. We determine the $\mathcal{P}/\mathcal{N}$-status for staircases and for $2$- and $3$-part partitions. We evaluate CRIM and its restrictions to certain partition families within the Conway-Gurvich-Ho classification scheme, establishing that CRIM is neither returnable nor domestic. We conjecture that every losing partition has even rank.

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Ina Bašić, Eric Gottlieb, Matjaž Krnc. 2026-06-15. CRIM: A Natural Game on Integer Partitions. https://arxiv.org/abs/2606.16828

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