## Restrictions of $m$-Wythoff Nim and $p$-complementary Beatty Sequences

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Larsson, Urban

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Fix a positive integer $m$. The game of \emph{$m$-Wythoff Nim} (A.S.
Fraenkel, 1982) is a well-known extension of \emph{Wythoff Nim}, a.k.a 'Corner
the Queen'. Its set of $P$-positions may be represented by a pair of increasing
sequences of non-negative integers. It is well-known that these sequences are
so-called \emph{complementary homogeneous}
\emph{Beatty sequences}, that is they satisfy Beatty's theorem. For a
positive integer $p$, we generalize the solution of $m$-Wythoff Nim to a pair
of \emph{$p$-complementary}---each positive integer occurs exactly $p$
times---homogeneous Beatty sequences $a = (a_n)_{n\in \M}$ and $b = (b_n)_{n\in
\M}$, which, for all $n$, satisfies $b_n - a_n = mn$. By the latter property,
we show that $a$ and $b$ are unique among \emph{all} pairs of non-decreasing
$p$-complementary sequences. We prove that such pairs can be partitioned into
$p$ pairs of complementary Beatty sequences. Our main results are that
$\{\{a_n,b_n\}\mid n\in \M\}$ represents the solution to three new
'$p$-restrictions' of $m$-Wythoff Nim---of which one has a \emph{blocking
maneuver} on the \emph{rook-type} options. C. Kimberling has shown that the
solution of Wythoff Nim satisfies the \emph{complementary equation}
$x_{x_n}=y_n - 1$. We generalize this formula to a certain '$p$-complementary
equation' satisfied by our pair $a$ and $b$. We also show that one may obtain
our new pair of sequences by three so-called \emph{Minimal EXclusive}
algorithms. We conclude with an Appendix by Aviezri Fraenkel.

Comment: 22 pages, 2 figures, Games of No Chance 4, Appendix by Aviezri Fraenkel

Comment: 22 pages, 2 figures, Games of No Chance 4, Appendix by Aviezri Fraenkel

##### Keywords

Mathematics - Combinatorics, 91A46