Variants of Wythoff's Games with Different Terminal Sets

TL;DR

This paper explores a variant of Wythoff's game with a modified terminal set, providing a non-recursive Fibonacci-based description of P-positions for the game.

Contribution

It introduces a new variant of Wythoff's game with a specific terminal set and characterizes its P-positions using Fibonacci numbers without recursion.

Findings

P-positions are described by Fibonacci sequence

The variant generalizes classical Wythoff's game

A non-recursive Fibonacci characterization is provided

Abstract

We study a variant of the classical Wythoff's game. The classical form is played with two piles of stones, from which two players take turns to remove stones from one or both piles. When removing stones from both piles, an equal number must be removed from each. The player who removes the last stone or stones is the winner. Equivalently, we consider a single chess queen placed somewhere on a large grid of squares. Each player can move the queen toward the upper-left corner of the grid, either vertically, horizontally, or diagonally in any number of steps. The winner is the player who moves the queen to the terminal position in the upper-left corner, the position (0,0) in our coordinate system. Let k be a positive integer, and we consider the variant of Wythoff's game with the terminal set {(x,y):x,y are non-negative integers and x+y <=k}. The set of P-positions of this variant is described by the Fibonacci sequence without using recursion.