A modification of Wythoff's Nim

TL;DR

This paper introduces a modified version of Wythoff's Nim with a new move called 'split' and explores its P-positions using Tribonacci and k-bonacci words, providing a new combinatorial analysis.

Contribution

It presents a novel modification of Wythoff's Nim incorporating 'split' moves and links the P-positions to Tribonacci words, extending to k-bonacci words and deriving a mex-rule for the Quadribonacci table.

Findings

P-positions are coded by the Tribonacci word

A mex-rule generates the Quadribonacci table

Analysis of k-bonacci words' letter positions

Abstract

We modify Wythoff's game by allowing an additional move, which we call a "split", and show how the $P$-positions are coded by the Tribonacci word. We analyze the table of letter positions of arbitrary $k$-bonacci words and find a $\mathrm{mex}$-rule that generates the Quadribonacci table.