Maharaja Nim, Wythoff's Queen meets the Knight

TL;DR

This paper introduces Maharaja Nim, a new combinatorial game extending Wythoff Nim with chess Queen and Knight moves, and proves the structure of winning positions is closely related to the golden ratio lines, also analyzing a variant with specific Knight jumps.

Contribution

The paper extends Wythoff Nim by adding Knight moves, analyzes the structure of winning positions, and demonstrates polynomial-time decision algorithms for certain variants.

Findings

Winning positions are within bounded distance of lines with slopes related to the golden ratio.

The game structure closely resembles Wythoff Nim's, with positions near specific lines.

Polynomial time complexity is established for a variant with (2,3) and (3,2) Knight jumps.

Abstract

New combinatorial games are introduced, of which the most pertinent is Maharaja Nim. The rules extend those of the well-known impartial game of Wythoff Nim in which two players take turn in moving a single Queen of Chess on a large board, attempting to be the first to put her in the lower left corner. Here, in addition to the classical rules a player may also move the Queen as the Knight of Chess moves. We prove that the second player's winning positions of Maharaja Nim are close to the ones of Wythoff Nim, namely they are within a bounded distance to the lines with slope $\frac{\sqrt{5}+1}{2}$ and $\frac{\sqrt{5}-1}{2}$ respectively. For a close relative to Maharaja Nim, where the Knight's jumps are of the form $(2,3)$ and $(3,2)$ (rather than $(1,2)$ and $(2,1)$), we also demonstrate polynomial time complexity to the decision problem of the outcome of a given position.