A rectair is the rectair family used in CRIM, and G denotes its Sprague–Grundy value. Near-square rectair conjecture. For r≥7, G(R_r,r-1^k)= 3& k=r-2 and r is odd, 1&otherwise. In addition, G(R^1_3,2)= G(R^3_5,4)=3,qquad G(R^2_4,3)=2,qquad G(R^4_6,5)=5. All other rectairs have Sprague–Grundy value 1. These values are proposed as an extension of the computed rectair cases; the source gives no resolution.

The Sprague–Grundy conjecture for near-square rectairs

Less than 1 year old · traced to 2026

A rectair is the rectair family used in CRIM, and $\mathcal G$ denotes its Sprague–Grundy value. Near-square rectair conjecture. For $r\geq7$ ,

\mathcal G(R_{r,r-1}^k)= \begin{cases} 3& k=r-2\text{ and }r\text{ is odd},\\ 1&\text{otherwise.} \end{cases}

In addition,

\mathcal G(R^1_{3,2})=\mathcal G(R^3_{5,4})=3,\qquad \mathcal G(R^2_{4,3})=2,\qquad \mathcal G(R^4_{6,5})=5.

All other rectairs have Sprague–Grundy value $1$ . These values are proposed as an extension of the computed rectair cases; the source gives no resolution.

References

Primary source

Ina Bašić, Eric Gottlieb and Matjaž Krnc, “CRIM: A Natural Game on Integer Partitions”, arXiv:2606.16828 (2026).

combinatorial-game-theory integer-partitions

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unverified posted calculation claims the conjecture is false in both parity cases, but no independent confirmation was found.

Bašić, Gottlieb, and Krnc proposed the near-square formula and its exceptional small cases in their June 2026 paper on CRIM. The paper presents these values as conjectural extensions of computed cases and does not claim a resolution.

Posted attempt

A reader-written calculation claims a complete disproof: it reports $\mathcal G(R^5_{7,6})=1$ rather than the predicted $3$ , and $\mathcal G(R^6_{8,7})=3$ rather than the predicted $1$ , using recursive mex computations. The attempt has not been independently verified.

Current status (as of August 2026): The conjectured formula is unproved, while an unverified calculation claims counterexamples at $r=7$ and $r=8$ ; the broader pattern therefore remains unsettled.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

The near-square rectair conjecture fails in both parity branches, already at the first two values in its stated range $r\ge7$ .

Write $\lambda'$ for the conjugate partition. The exact normal-play recursion is

G(\varnothing)=0,\qquad G(\lambda) = \operatorname{mex}\!\left( \{G(\lambda\text{ with one row deleted})\} \cup \{G((\lambda'\text{ with one row deleted})')\} \right).

Every recursive call strictly decreases the number of boxes.

For $r=7$ and $k=5$ , the source's rectair definition gives

R^5_{7,6}=(6,6,5,4,3,2,1).

Its distinct followers and exact recursively computed Grundy values are

\begin{array}{c|c} \text{follower}&G\\ \hline (6,6,5,4,3,2)&0\\ (6,6,5,4,3,1)&2\\ (6,6,5,4,2,1)&0\\ (6,6,5,3,2,1)&4\\ (6,6,4,3,2,1)&0\\ (6,5,4,3,2,1)&0\\ (5,5,5,4,3,2,1)&2\\ (5,5,4,4,3,2,1)&4\\ (5,5,4,3,3,2,1)&0\\ (5,5,4,3,2,2,1)&4\\ (5,5,4,3,2,1,1)&0\\ (5,5,4,3,2,1)&5 \end{array}

Consequently

G(R^5_{7,6}) = \operatorname{mex}\{0,2,4,5\} = 1,

whereas the conjecture predicts $3$ , because $k=r-2$ and $r$ is odd.

Conversely, take $r=8$ and $k=6$ . For

R^6_{8,7}=(7,7,6,5,4,3,2,1),

the fourteen distinct followers have Grundy values

0,0,0,0,0,0,0,1,2,2,2,2,2,4.

Therefore

G(R^6_{8,7}) = \operatorname{mex}\{0,1,2,4\} = 3,

whereas the conjecture predicts $1$ .

The parity error is also exposed by the source's own subsequent conjecture. Its padded staircase satisfies

\mathrm{PS}_n=R^{n-1}_{n+1,n}.

Thus Conjecture 3 predicts $G(\mathrm{PS}_n)=3$ for even $n\ge6$ and $1$ for odd $n\ge7$ , while Conjecture 5 predicts exactly the opposite on both infinite families. The exact cases $n=6,7$ identify Conjecture 3 as false. The suggested parity repair for all larger $r$ remains a separate open claim.

Source: I. Bašić, E. Gottlieb, and M. Krnc, “CRIM: A Natural Game on Integer Partitions,” arXiv:2606.16828, Conjectures 3 and 5.