A New Twist on Wythoff's Game

TL;DR

This paper introduces Twyst-off, a novel multi-stack extension of Wythoff's Game inspired by knot theory, analyzing P-positions, symmetry, and structural properties for finite and infinite stacks.

Contribution

It defines Twyst-off, proves existence and uniqueness of three-stack P-positions, and explores structural and symmetry properties for multiple stacks, including infinite stacks.

Findings

Many three-stack P-positions are symmetric.

Established bounds on non-symmetric three-stack P-positions.

Classified positions with infinite stacks up to six stacks.

Abstract

Wythoff's Game is a game for two players playing alternately on two stacks of tiles. On her turn, a player can either remove a positive number of tiles from one stack, or remove an equal positive number of tiles from both stacks. The last player to move legally wins the game. We propose and study a new extension of this game to more than two stacks, which we call Twyst-off, inspired by the Reidemeister moves of knot theory. From an ordered sequence of stacks of tiles, a player may either remove a positive number of tiles from one of the two end stacks, or remove the same positive number of tiles from two consecutive stacks. Whenever an interior stack is reduced to 0, the two neighboring stacks are combined. In this paper, we prove several results about those Twyst-off positions that can be won by the second player (these are called P-positions). We prove an existence and uniqueness result that makes the visualization of data on three-stack P-positions possible. This shows that many such positions are symmetric, like the easy general examples (a,a,a) and (a,a+1,a). The main result establishes tight bounds on those three-stack P-positions that are not symmetric. We go on to prove one general structural result for positions with an arbitrary number of stacks. We also prove facts about the game when allowing stacks of infinite size, including classifying all positions with only infinite stacks in sequences of up to six stacks.