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

TL;DR

This paper introduces switch operators for combinatorial games, allowing players to change rules mid-game, and analyzes their properties and specific cases involving Nim, Wythoff, Euclid, and Domineering variants.

Contribution

It defines and studies push-the-button games with switch operators, extending classical combinatorial game theory to new rule-changing scenarios.

Findings

Compatibility of rulesets ensures well-defined switch games.

Periodicity results for Subtraction games extend to switch operators.

Partial results obtained for a Domineering variation with horizontal and vertical placements.

Abstract

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 (R 1 , R 2) is compatible if, given any position in R 1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2), we build the push-the-button game R 1 R 2 , where players start by playing according to the rules R 1 , but at some point during play, one of the players must switch the rules to R 2 , by pushing the button ". Thus, the game ends according to the terminal condition of ruleset R 2. 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 R 1 is the game where the players put the domino tiles horizontally and R 2 the game where they play vertically (thus generalizing the octal game 0.07).