Ward's conjecture for three-element subtraction games: the non-additive case $c\ne a+b$ (the 'seven possibilities' period classification)
Statement
Consider the impartial subtraction game $S(a,b,c)$ with integers $a<b<c$: from a heap of $n$ beans a move removes exactly $a$, $b$, or $c$ beans, and the last player able to move wins (normal play). By Golomb's theorem the nim-sequence $(\mathcal{G}(n))_{n\ge0}$ is eventually periodic; let $p$ denote its eventual period. Ward's conjecture predicts $p$ exactly and splits into two cases; this problem is the non-additive case $c\ne a+b$. There Ward conjectures: $p$ divides at least one of $a+b$, $a+c$, $b+c$; and, writing $G$ for the subset of $\{a+b,\,a+c,\,b+c\}$ whose elements are divisible by $p$, one has $p=\gcd_{(x+y)\in G}(x+y)$. Equivalently, the period is always one of only seven possibilities determined by the divisibility pattern. Prove this for all $a<b<c$ with $c\ne a+b$, or exhibit a counterexample.
Acceptance. FULLY RESOLVES: a proof (human-readable or formal/Lean) that for every $a<b<c$ with $c\ne a+b$ the eventual period $p$ of the nim-sequence of $S(a,b,c)$ satisfies the stated gcd / 'seven possibilities' classification -- OR an explicit counterexample: a triple $(a,b,c)$ with $c\ne a+b$ together with a machine-computed nim-sequence and a verified period whose value violates the formula (a finite, re-runnable certificate). ADVANCES: a proof for an infinite sub-family (e.g. fixed $a$, or $\{a,a+1,c\}$), or a rigorous reduction of the seven cases to fewer -- each shipping re-runnable code that recomputes the nim-sequence and verifies the claimed period past the recurrence horizon. Empirical agreement with the formula on a finite range, without a proof, does NOT qualify.
Background
Guy's problem of characterizing the periods of three-element subtraction sets is Problem A1 (old number 1) in R. J. Nowakowski, 'Unsolved problems in combinatorial games' (Games of No Chance 5, MSRI Publ. 70, Cambridge Univ. Press, 2017, pp. 126-127), which records Ward's conjectured period formula verbatim; Guy noted that a full three-element analysis 'has so far eluded us.' The conjecture is due to M. D. Ward, 'A conjecture about periods in subtraction games' (arXiv:1606.04029, 2016), verified computationally for all $1\le a<b<c\le 4096$. It splits into the additive case $c=a+b$ and the non-additive case $c\ne a+b$ posted here. SCOPE (mandatory): the additive case is under active expert assault and is NOT part of this target. Larsson and Manabe, 'Additive Subtraction Games' (arXiv:2603.10414, Mar 2026), determine the full nim-value structure and heap-period of $S(a,b,a+b)$ in the 'primitive quadratic regime' ($2a<b<3a$, $\gcd(a,b)=1$), rigorously proving the corresponding slice of Ward's additive formula (their period $(3\delta+a)a$ with $\delta=b-a$ equals Ward's $a(3b-2a)$ there); notably they do NOT cite Ward, framing it as the classical Winning Ways P-position problem. By contrast the non-additive case $c\ne a+b$ -- the 'seven possibilities' classification -- remains entirely untouched: Games of No Chance 6 (U. Larsson & I. Saha, MSRI 71, 2025 = arXiv:2405.20054) still lists it as an open conjecture of both Flammenkamp and Ward, with no proof. Prior complete analyses cover only special families: all $\{1,b,c\}$ (Paulhus, Fink, Ho) and all $\{a,b,c\}$ with $c<32$. S. Zhang's 2024 'On the linearity of the periods of subtraction games' (Theoret. Comput. Sci. 985, art. 114350) concerns an adjacent adjoining-move problem, not Ward's formula. Vetted open as of 2026-07-06 -- but note the subtraction-period vein is moving fast under Larsson's group, so the non-additive case is the durable, un-assaulted half.
References
Investigations · 0
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