19,057 research outputs found
Restrictions of -Wythoff Nim and -complementary Beatty Sequences
Fix a positive integer . The game of \emph{-Wythoff Nim} (A.S.
Fraenkel, 1982) is a well-known extension of \emph{Wythoff Nim}, a.k.a 'Corner
the Queen'. Its set of -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 , we generalize the solution of -Wythoff Nim to a pair
of \emph{-complementary}---each positive integer occurs exactly
times---homogeneous Beatty sequences a = (a_n)_{n\in \M} and b = (b_n)_{n\in
\M}, which, for all , satisfies . By the latter property,
we show that and are unique among \emph{all} pairs of non-decreasing
-complementary sequences. We prove that such pairs can be partitioned into
pairs of complementary Beatty sequences. Our main results are that
\{\{a_n,b_n\}\mid n\in \M\} represents the solution to three new
'-restrictions' of -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}
. We generalize this formula to a certain '-complementary
equation' satisfied by our pair and . 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
Fraenke
Partition games
We introduce CUT, the class of 2-player partition games. These are NIM type
games, played on a finite number of heaps of beans. The rules are given by a
set of positive integers, which specifies the number of allowed splits a player
can perform on a single heap. In normal play, the player with the last move
wins, and the famous Sprague-Grundy theory provides a solution. We prove that
several rulesets have a periodic or an arithmetic periodic Sprague-Grundy
sequence (i.e. they can be partitioned into a finite number of arithmetic
progressions of the same common difference). This is achieved directly for some
infinite classes of games, and moreover we develop a computational testing
condition, demonstrated to solve a variety of additional games. Similar results
have previously appeared for various classes of games of take-and-break, for
example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and
Guy (1982). In this context, our contribution consists of a systematic study of
the subclass `break-without-take'
On Aperiodic Subtraction Games with Bounded Nim Sequence
Subtraction games are a class of impartial combinatorial games whose
positions correspond to nonnegative integers and whose moves correspond to
subtracting one of a fixed set of numbers from the current position. Though
they are easy to define, sub- traction games have proven difficult to analyze.
In particular, few general results about their Sprague-Grundy values are known.
In this paper, we construct an example of a subtraction game whose sequence of
Sprague-Grundy values is ternary and aperiodic, and we develop a theory that
might lead to a generalization of our construction.Comment: 45 page
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