A Generalized Diagonal Wythoff Nim

Preprint, 2010

In this paper we study a family of 2-pile take away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN).The story begins with 2-pile Nim whose sets of moves and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and $\{(t,t)\mid t\in \M \}$ respectively. If we adjoin to 2-pile Nim the main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as moves, the new game is Wythoff Nim. It is well-known that the $P$-positions of this game lie on two 'beams' starting at the origin with slopes $\phi = \frac{1+\sqrt{5}}{2}>1$ and $\frac{1}{\phi } < 1$. Hence one may think of this as if, in the process of going from Nim to Wythoff Nim, the set of $P$-positions has \emph{split} and landed some distance off the main diagonal. This geometrical observation has motivated us to ask the following intuitive question. Does this splitting of the set of $P$-positions continue in some meaningful way if we adjoin to the game of Wythoff Nim new \emph{generalized diagonal} moves of the form $(pt, qt)$ and $(qt, pt)$, where $p < q$ are fixed positive integers and $t$ ranges over the positive integers? Does the answer depend on the specific values of $p$ and $q$? We state three conjectures of which the weakest form is: $\lim_{t\in \N}\frac{b_t}{a_t}$ exists, and equals $\phi$, if and only if $(p, q)$ is a certain \emph{non-splitting pair}, and where $\{(a_t, b_t),(b_t,a_t)\}$ represents the set of $P$-positions of the new game. Then we prove this conjecture for the special case $(p,q) = (1,2)$ (a \emph{splitting pair}). We prove the other direction whenever $q / p < \phi$. A variety of experimental data is included, aiming to point out some directions for future work on GDWN games.