A Generalized Diagonal Wythoff Nim

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 $\{\{0,t\}\mid t\in \N\}$ and \{(t,t)\mid t\in \M \} respectively. If we to 2-pile Nim adjoin the main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ 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: $\lim_{t\in \N}\frac{b_t}{a_t}$ 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.Comment: 38 pages, 34 figure

The ⋆\star-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is \W^\star= (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a polynomial time algorithm for infinitely many $P$-positions of \W^\star. A repeated application of $\star$ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the \emph{permutation games}, which we introduce here. We also introduce the family of \emph{ornament games}, whose $P$-positions define complementary Beatty sequences with rational moduli---hence related to A. S. Fraenkel's `variant' Rat- and Mouse games---and give closed forms for the moves of such games. We also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.Comment: 30 pages, 5 figure