A Generalized Diagonal Wythoff Nim

Abstract

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 options and $P$-positions are $\{\{0,t\}\mid t\in \mathbb{N}\}$ and $\{(t,t)\mid t\in \M \}$ respectively. If we to 2-pile Nim adjoin the main-\emph{diagonal} $\{(t,t)\mid t\in \mathbb{N}\}$ as options, the new game is Wythoff Nim. It is well-known that the $P$-positions of this game lie on two 'beams' originating 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, to the game of Wythoff Nim, adjoin some new \emph{generalized diagonal} move, that is a move of the form $\{pt,qt\}$, where $0<p<q$ are fixed positive integers and $t>0$? Does the answer perhaps depend on the specific values of $p$ and $q$? We state three conjectures of which the weakest form is: ${\lim }_{t\in \mathbb{N}}\frac{b_t}{a_t}$ exists, and equals $\Phi$, if and only if $(p,q)$ is a certain \emph{non-splitting pair}, and where $\{\{a_t,b_t\}\}$ 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 $q/p<\Phi$. In the Appendix, a variety of experimental data is included, aiming to point out some directions for future work on GDWN games.

38 pages, 34 figures

References in corpus (1)

• 2-pile Nim with a Restricted Number of Move-size Imitations