A Weak Solution of Inverse Treblecross

Abstract

We study Inverse Treblecross, the impartial avoidance version of the game Treblecross, in which completing a line of three X's is prohibited. We give a weak solution of the game: an empty board of $n$ cells is a $\mathcal{P}$-position (Sprague--Grundy value $0$) if and only if $n\equiv 0,2,3,6,9(\mathrm{mod}10)$, and has value $1$ if $n\equiv 1,4,5,7,8(\mathrm{mod}10)$. In particular the empty-board Sprague--Grundy sequence is the purely periodic sequence of period $10$ with values in $\{0,1\}$. The proof introduces two characteristic functions on $\mathbb{Z}/10\mathbb{Z}$ and a family of regular positions, and shows by induction that such a position splits, under a reduction move, into a disjunctive sum of regular positions whose sign product is preserved or inverted.