Unique Winning Opening Move in Three-Row Chomp

Abstract

Chomp was introduced by Gale in 1974 \cite{Gale1974}. In the same paper, Gale reported that the $3\times n$ games had been completely analyzed for $n\le 100$, with a unique winning first move in every case, and asked whether winning first moves are unique in general. Although the general uniqueness statement is false \cite[Section~7.1]{BrouwerEtAl2005}, we prove that the three-row uniqueness phenomenon suggested by Gale&#39;s computations holds for all $n$: every $3\times n$ Chomp rectangle has exactly one winning opening move. This settles the three-row case of Gale&#39;s 52-year-old first-move uniqueness question. The proof is carried out in the two-variable recurrence introduced by Brouwer, Horváth, Molnár-Sáska, and Szabó \cite{BrouwerEtAl2005} for the function $f(q,r)$ whose values encode the $P$-positions. The main local ingredient is a rightmost-hole principle: if a value $p$ is absent from the set $C(q,r)$ but belongs to all corresponding sets $C(t,r)$ for $q<t<p$, then all intermediate values $q+1,\ldots,p-1$ are forced to belong to $C(q,r)$. This separates the diagonal values from the starts of constant rows, and yields a partition of the positive integers into the two possible types of winning opening moves. It also identifies the row of the unique opening move: no first-row opening move is winning; the second-row and third-row cases are precisely the two complementary Chomp sequences A029900 and A029901.