When are translations of P-positions of Wythoff's game P-positions?

We study the problem whether there exist variants of {\sc Wythoff}'s game whose $\P$-positions, except for a finite number, are obtained from those of {\sc Wythoff}'s game by adding a constant $k$ to each $\P$-position. We solve this question by introducing a class \{\W_k\}_{k \geq 0} of variants of {\sc Wythoff}'s game in which, for any fixed $k \geq 0$, the $\P$-positions of \W_k form the set $\{(i,i) | 0 \leq i < k\}\cup \{(\lfloor \phi n \rfloor + k, \lfloor \phi^2 n \rfloor + k) | n\ge 0\}$, where $\phi$ is the golden ratio. We then analyze a class \{\T_k\}_{k \geq 0} of variants of {\sc Wythoff}'s game whose members share the same $\P$-positions set $\{(0,0)\}\cup \{(\lfloor \phi n \rfloor + 1, \lfloor \phi^2 n \rfloor + 1) | n \geq 0 \}$. We establish several results for the Sprague-Grundy function of these two families. On the way we exhibit a family of games with different rule sets that share the same set of $\P$-positions

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

Wythoff Wisdom

International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems

Impartial Games and Recursive Functions

Interest in 2-player impartial games often concerns the famous theory of Sprague-Grundy. In this thesis we study other aspects, bridging some gaps between combinatorial number theory, computer science and combinatorial games. The family of heap games is rewarding from the point of view of combinatorial number theory, partly because both the positions and the moves are represented simply by finite vectors of nonnegative integers. For example the famous game of Wythoff Nim on two heaps of tokens has a solution originating in Beatty sequences with modulus the Golden ratio. Sometimes generalizations of this game have similar properties, but mostly they are much harder to grasp fully. We study a spectrum of such variations, and our understanding of them ranges from being complete in the case of easier problems, to being very basic in the case of the harder ones. One of the most far reaching results concerns the convergence properties of a certain $\star\star$-operator for invariant subtraction games, introduced here to resolve an open problem in the area. The convergence holds for any game in any finite dimension. We also have a complete understanding of the reflexive properties of such games. Furthermore, interesting problems regarding computability can be formulated in this setting. In fact, we present two Turing complete families of impartial (heap) games. This implies that certain questions regarding their behavior are algorithmically undecidable, such as: Does a given finite sequence of move options alternate between N- and P-positions? Do two games have the same sets of P-positions? The notion of N- and P-positions is very central to the class of normal play impartial games. A position is in P if and only if it is safe to move there. This is virtually the only theory that we need. Therefore we hope that our material will inspire even advanced undergraduate students in future research projects. However we would not consider it impossible that the universality of our games will bridge even more gaps to other territories of mathematics and perhaps other sciences as well. In addition, some of our findings may apply as recreational games/mathematics

Some notes on impartial games and NIM dimension

Tese de doutoramento, Matemática (Análise Numérica e Matemática Computacional), Universidade de Lisboa, Faculdade de Ciências, 2010.Disponível no documento

Finite subtraction games in more than one dimension

We study two-player impartial vector subtraction games (on tuples of nonnegative integers) with finite rulesets, and solve all two-move games. Through multiple computer visualizations of outcomes of two-dimensional rulesets, we observe that they tend to partition the game board into periodic mosaics on very few regions/segments, which can depend on the number of moves in a ruleset. For example, we have found a five-move ruleset with an outcome segmentation into six semi-infinite slices. We prove that games in two dimensions are row/column eventually periodic. Several regularity conjectures are provided. Through visualizations of some rulesets, we pose open problems on the generic hardness of games in two dimensions.Comment: 38 page