Rulesets for Beatty games

Abstract

We describe a ruleset for a 2-pile subtraction game with P-positions $$\{(\lfloor \alpha n \rfloor ,\lfloor \beta n \rfloor ) : n \in \mathbb Z_{\ge 0} \}$$ for any irrational $$1< \alpha < 2$$ , and $$\beta $$ such that $$1/\alpha +1/\beta = 1$$ . We determine the $$\alpha $$ ’s for which the game can be represented as a finite modification of t-Wythoff (Holladay, Math Mag 41:7–13, 1968; Fraenkel, Am Math Mon 89(6):353–361, 1982) and describe this modification.