Summary: 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 $0 \le x \le y$, the outcome of the $upper$ position $(x, y)$ is identical to that of $(y, x)$. First we prove that $\phi ^{-1} = 2/(1+\sqrt 5)$ 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.

combinatorial game, complementary sequence, golden ratio, impartial game, integer sequence, lower asymptotic density, splitting sequence, Wythoff nim