Nim with pass, problem A13

Given a Nim position $\mathbf{x}=(x_1,\ldots,x_n)\in\mathbb{Z}_{\ge 0}^n$ together with a pass-status $\varepsilon\in\{0,1\}$ , where $\varepsilon=1$ means that one pass remains available, a move either replaces one nonzero heap $x_i$ by an integer $y$ with $0\le y<x_i$ , leaving $\varepsilon$ unchanged, or, when $\varepsilon=1$ , changes the pass-status to $0$ without changing the heaps. The problem is to give a complete characterization of the winning and losing positions, equivalently to determine exactly which states $(\mathbf{x},\varepsilon)$ are $\mathcal{P}$ -positions.

Progress summary

Partially solved

New structural results clarify some cases, but Nim with a pass has not been solved in general.

The problem seeks a general description of winning and losing positions in Nim after allowing one pass. Earlier work explicitly left an analogous solution open, and no complete proof or counterexample has been reported.

Known results

Three or more piles: the Nimber Extension Rule gives limited progress, but the relevant structure remains unresolved (2020).
Restricted component games: the pass-game value reduces to Nim with a pass on the components' ordinary Grundy values (2025).
For positions $(x,y,1)$ , when $x \ge 9$ or $y \ge 9$ , the position is a $\mathcal{P}$ -position exactly when $(x,y,0)$ has Grundy number $1$ (2025).

August 2026 split-sum results

Recent work introduces a generalized split-sum and Grundy-value framework, identifying cases where adding a pass behaves like an ordinary disjunctive sum and deriving further results for selected Nim configurations. It presents structural progress, not a solution of A13.

Current status (as of August 2026): Partial structural results are established, but the general Nim-with-pass problem remains open.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Some results on split sum of impartial games — arXiv — François Carret

combinatorial-game-theory

3 days ago