Wythoff Nim Extensions and Splitting Sequences
Chalmers University of Technology
and University of Gothenburg
We study extensions of the classical impartial combinatorial game of
Wythoff Nim. The games are played on two heaps of tokens, and have
symmetric move options, so that, for any integers 0 ≤ x
the outcome of the upper position
(x, y) is identical to that
of (y, x).
First we prove that φ-1 = 2/(1+√5) is a
lower bound for the lower asymptotic density of the x-coordinates of
a given game's upper P-positions. The second result concerns a
subfamily, called a Generalized Diagonal Wythoff Nim, recently
introduced by Larsson. A certain split of P-positions,
distributed in a number of so-called P-beams, was conjectured for many
such games. The term split here means that an infinite sector of
upper positions is void of P-positions, but with infinitely many upper
P-positions above and below it. By using the first result, we prove
this conjecture for one of these games, called (1,2)-GDWN, where a
player moves as in Wythoff Nim, or instead chooses to remove a positive
number of tokens from one heap and twice that number from the other.
Full version: pdf,
Received September 18 2012;
revised versions received November 22 2013; January 6 2014; February 18 2014;
March 10 2014; April 3 2014.
Published in Journal of Integer Sequences, April 4 2014.
Journal of Integer Sequences home page