4 research outputs found
Heap games, numeration systems and sequences
We propose and analyse a 2-parameter family of 2-player games on two heaps of
tokens, and present a strategy based on a class of sequences. The strategy
looks easy, but is actually hard. A class of exotic numeration systems is then
used, which enables us to decide whether the family has an efficient strategy
or not. We introduce yet another class of sequences, and demonstrate its
equivalence with the class of sequences defined for the strategy of our games.Comment: To appear in Annals of Combinatoric
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
Counting Minimal Semi-Sturmian Words
A finite Sturmian word w is a balanced word over the binary alphabet {a,b}, that is, for all subwords u andv of w of equal length, ||u|a-|v|a|=1, where |u|a and |v|a denote the number of occurrences of the lettera in u and v, respectively. There are several other characterizations, some leading to efficient algorithms for testing whether a finite word is Sturmian. These algorithms find important applications in areas such as pattern recognition, image processing, and computer graphics. Recently, Blanchet-Sadri and Lensmire considered finite semi-Sturmian words of minimal length and provided an algorithm for generating all of them using techniques from graph theory. In this paper, we exploit their approach in order to count the number of minimal semi-Sturmian words. We also present some other results that come from applying this graph theoretical framework to subword complexity
Wythoff Wisdom
International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems