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.

Author

Urban Larsson

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Subject Categories

Discrete Mathematics

More information

Created

10/7/2017