The $\star$-operator and Invariant Subtraction Games

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 $\mathbf{0}$. 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 $\mathbf{0}$, as moves in the new game. One such game is ${\text{W}}^{\star }=$ (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 ${\text{W}}^{\star }$. 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.