arXiv subjects
Publications and source records attributed to Matthieu Dufour.
We define and give results on the game NecklaceNim NN($n$,$k$) which is PathNim PN($n$,$k$) with an additional move allowed on the end vertices. This game arises as a sub-game in the context of solving CircularNim CN($n$,$k$) when $k-2$ consecutive stacks have been depleted, therefore its solution is critical to solving CircularNim. We solve the infinite families of NN($n$,$k$) when play is allowed on at least half the stacks.
We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. This process is suitable for all variations of the game Nim whose rule sets form a simplicial complex. We rephrase Simplicial Nim as Set Nim SN($n,A$) and derive results on the structure of the P-positions in terms of invariant vectors, without needing the background and notation of simplicial complexes. We also show that invariant vectors differ from the circuits used to describe the P-positions in Simplicial Nim and that invariant vectors have wider applicability compared to circuits. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN($n,k$) with $n=7, k=3$ and $n=8,k=3$.
We introduce the game Slow $A$-Nim which generalizes a number of recently studied games. Slow $A$-Nim is played on $n$ stacks of tokens, and the set $A$ indicates the number of stacks a player can play on. Once a player has decided on the number $a$ of stacks, s/he will select any $a$ stacks and then remove one token from each stack. The last player to move wins. We give results on the $\mathcal{P}$-positions of Slow $A$-Nim for several infinite families. The results for $A = \{n-1\}$, which is the game Slow Exact $k$-Nim for $k=n-1$ extend recent results for small values of $n$. The other two families, $A=\{n-1,n\}$ and $A=\{1,n\}$ have not been previously studied. The $\mathcal{P}$-positions for $A = \{n-1\}$ and $A = \{n-1,n\}$ are closely related and have a very elegant description in terms of reduced positions, that is, positions for which unplayable tokens are disregarded. We also provide some general results that will be useful in the study of other sets $A$.
Circular Nim is a two-player impartial combinatorial game consisting of $n$ stacks of tokens placed in a circle. A move consists of choosing $k$ consecutive stacks and taking at least one token from one or more of the stacks. The last player able to make a move wins. The question of interest is: Who can win from a given position if both players play optimally? In an impartial combinatorial game, there are only two types of positions. An $\mathcal{N}$-position is one from which the next player to move has a winning strategy. A $\mathcal{P}$-position is one from which the next player is bound to lose, no matter what moves s/he makes. Therefore, the question who wins is answered by identifying the $\mathcal{P}$-positions. We will prove results on the structure of the $\mathcal{P}$-positions for $n = 7$ and $k = 4$, extending known results for other games in this family. The interesting feature of the set of $\mathcal{P}$-positions of this game is that it splits into different subsets, unlike the structure for the known games in this family.
The main objective of this paper consists in creating a new class of copulae from various joint distributions occurring in connection with certain Brownian motion processes. We focus our attention on the distributions of univariate Brownian motions having a drift parameter and their maxima and on correlated bivariate Brownian motions by considering the maximum value of one of them. The copulae generated therefrom and their associated density functions are explicitly given as well as graphically represented.
We study the $\star$-operator (Larsson et al. 2011) of impartial vector subtraction games (Golomb 1965). Here we extend the notion to the misère-play convention, and prove convergence and other properties; notably more structure is obtained under misère-play as compared with the normal-play convention (Larsson 2012).
The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus belonging to the same family of games as e.g. nine-men's morris): first building, where players alternate to put one token on one of the, initially empty, stacks until all tokens have been used. Then, the players play nim. Of course, because the solution for the game of nim is known, the goal of the player who starts nim play is a placement of the tokens so that the Nim-sum of the stack heights at the end of building is different from 0. This game is trivial if the total number of tokens is odd as the Nim-sum could never be 0, or if both the number of tokens and the number of stacks are even, since a simple mimicking strategy results in a Nim-sum of 0 after each of the second player's moves. We present the solution for this game for some non-trivial cases and state a general conjecture.
A circular Nim game is a two player impartial combinatorial game consisting of n stacks of tokens placed in a circle. A move consists of choosing k consecutive stacks, and taking at least one token from one or more of the k stacks. The last player able to make a move wins. We prove results on the structure of the losing positions for small n and k and pose some open questions for further investigations.