Journal of Integer Sequences, Vol. 17 (2014), Article 14.5.7

Wythoff Nim Extensions and Splitting Sequences

Urban Larsson
Mathematical Sciences
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 ≤ xy, 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,    dvi,    ps,    latex    

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.

Return to Journal of Integer Sequences home page