Simple Chopsticks: Playing with any number of hands and fingers

TL;DR

This paper generalizes the game of Chopsticks to any number of fingers and hands, analyzing strategic advantages and fully characterizing outcomes for the two-finger case.

Contribution

It introduces a generalized version of Chopsticks with arbitrary hand and finger counts and provides a complete analysis for the two-finger scenario.

Findings

Having more hands than the opponent is advantageous.

The outcomes for two-finger hands are fully characterized.

The game dynamics change with increased number of hands.

Abstract

Chopsticks is a game played by two players where they start with one finger raised on each hand. On their turn, each player moves by pointing an attacking hand at one of their opponent's hands. The number of fingers on the pointed hand increases by the number of fingers on the attacking hand. If, after a move, a hand contains more than five fingers, it is removed from play. There are also other rules that allow players to move fingers from one hand to another, but we focus on this simple setup. We introduce a generalization of Chopsticks, called Simple Chopsticks, in which the players may have any number of $n$-fingered hands. We find that having more hands than your opponent is generally good, and use this fact to fully characterize the outcomes of \octopus/ in the case where the players have 2-fingered hands.