# Wythoff nim extensions and splitting sequences

**Journal of Integer Sequences**(1530-7638). Vol. 17 (2014), 5, p. artikel 14.5.7.

[Artikel, refereegranskad vetenskaplig]

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.

**Nyckelord: **Combinatorial game, Complementary sequence, Golden ratio, Impartial game, Integer sequence, Lower asymptotic density, Splitting sequence, Wythoff nim

Denna post skapades 2015-08-25. Senast ändrad 2016-08-22.

CPL Pubid: 221159