The ⋆-operator and Invariant Subtraction Games

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We study 2-player impartial games, so called 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 0. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, ⋆ of the old game, by taking the P-positions, except 0, as moves in the new game. One such game is ⋆= (Wythoff Nim)⋆, 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 ⋆. A repeated application of ⋆ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the permutation games, which we introduce here. We also introduce the family of 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)⋆⋆ = k-pile Nim.