When are translations of P-positions of Wythoff's game P-positions?

Abstract

We study the problem whether there exist variants of {\sc Wythoff}'s game whose $\P$-positions, except for a finite number, are obtained from those of {\sc Wythoff}'s game by adding a constant $k$ to each $\P$-position. We solve this question by introducing a class $\{\W_k\}_{k \geq 0}$ of variants of {\sc Wythoff}'s game in which, for any fixed $k\ge 0$, the $\P$-positions of $\W_k$ form the set $\{(i,i)|0\le i<k\}\cup \{(\lfloor \varphi n\rfloor +k,\lfloor {\varphi }^{2}n\rfloor +k)|n\ge 0\}$, where $\varphi$ is the golden ratio. We then analyze a class $\{\T_k\}_{k \geq 0}$ of variants of {\sc Wythoff}'s game whose members share the same $\P$-positions set $\{(0,0)\}\cup \{(\lfloor \varphi n\rfloor +1,\lfloor {\varphi }^{2}n\rfloor +1)|n\ge 0\}$. We establish several results for the Sprague-Grundy function of these two families. On the way we exhibit a family of games with different rule sets that share the same set of $\P$-positions.