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Wythoff Nim Extensions and Splitting Sequences
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Urban Larsson

Mathematical Sciences

Chalmers University of Technology

and University of Gothenburg

Göteborg

Sweden

**Abstract:**

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*
≤ *y*,
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.

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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.

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