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Larsson, U. (2010) *The $\star$-operator and Invariant Subtraction Games*.

** BibTeX **

@unpublished{

Larsson2010,

author={Larsson, Urban},

title={The $\star$-operator and Invariant Subtraction Games},

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.},

year={2010},

note={29},

}

** RefWorks **

RT Unpublished Material

SR Electronic

ID 134967

A1 Larsson, Urban

T1 The $\star$-operator and Invariant Subtraction Games

YR 2010

AB 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.

LA eng

LK http://urbanlarsson.mine.nu/homepage/preprints/StarZepe.pdf

OL 30