The switch operators and push-the-button games: A sequential compound over rulesets

We study operators that combine combinatorial games. This field was initiated by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R1,R2) is compatible if, given any position in R1, there is a description of how to move in R2. Given compatible (R1,R2), we build the push-the-button game R1⊚R2, where players start by playing according to the rules R1, but at some point during play, one of the players must switch the rules to R2, by pushing the button '⊚'. Thus, the game ends according to the terminal condition of ruleset R2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R1 is the game where the players put the domino tiles horizontally and R2 the game where they play vertically (thus generalizing the octal game 0.07).