The $\star$-operator and Invariant Subtraction Games

TL;DR

This paper explores invariant subtraction games, introduces the $oldsymbol{ ext{star}}$-operator, and analyzes properties of resulting games like Wythoff Nim, permutation games, and ornament games, providing algorithms and closed-form solutions.

Contribution

It introduces the $oldsymbol{ ext{star}}$-operator for invariant subtraction games, studies its effects, and characterizes special families like permutation and ornament games with new properties and solutions.

Findings

Polynomial time algorithm for infinitely many $P$-positions of $ ext{W}^ ext{star}$

Repeated $oldsymbol{ ext{star}}$-application yields well-structured permutation games

($k$-pile Nim)$^{ ext{star} ext{star}}$ equals $k$-pile Nim

Abstract

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is $\W^\star=$ (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a polynomial time algorithm for infinitely many $P$-positions of $\W^\star$. A repeated application of $\star$ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the \emph{permutation games}, which we introduce here. We also introduce the family of \emph{ornament games}, whose $P$-positions define complementary Beatty sequences with rational moduli---hence related to A. S. Fraenkel's `variant' Rat- and Mouse games---and give closed forms for the moves of such games. We also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.