相关论文: Mis\`ere-play Hackenbush Sprigs
We describe PNim and RNim, two variants of Nim in which piles of tokens are replaced with integer partitions or hyperrectangles. In PNim, the players choose one of the integer partitions and remove a positive number of rows or a positive…
We consider a random walk in a truncated cone $K_N$, which is obtained by slicing cone $K$ by a hyperplane at a growing level of order $N$. We study the behaviour of the Green function in this truncated cone as $N$ increases. Using these…
A Subtraction-Division game is a two player combinatorial game with three parameters: a set S, a set D, and a number n. The game starts at n, and is a race to say the number 1. Each player, on their turn, can either move the total to n-s…
In this paper, we will be proving mathematically that scoring play combinatorial game theory covers all combinatorial games. That is, there is a sub-set of scoring play games that are identical to the set of normal play games, and a…
In combinatorial game theory, there are two famous winning conventions, normal play and mis\`ere play. Under normal play convention, the winner is the player who moves last and under mis\`ere play convention, the loser is the player who…
We investigate the Sprague-Grundy sequences for two normal-play impartial games based on arithmetic functions, first described by Iannucci and Larsson in \cite{sum}. In each game, the set of positions is N (natural numbers). In saliquant,…
We propose a variant of Nim, named StrNim. Whereas a position in Nim is a tuple of non-negative integers, that in StrNim is a string, a sequence of characters. In every turn, each player shrinks the string, by removing a substring repeating…
For any partition of a positive integer we consider the chess (or draughts) colouring of its associated Ferrers graph. Let b denote the total number of black unit squares, and w the number of white squares. In this note we characterize all…
We present take-away games whose Sprague-Grundy functions are given by the Nim sum of heap sizes in a mixed base $\beta$. Let $\Delta_\beta$ be the set of such games. We give a necessary and sufficient condition for the existence of a…
We define a family of vertex colouring games played over a pair of graphs or digraphs $(G,H)$ by players $\forall$ and $\exists$. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is…
The ordinary game of Nim has a long history and is well-known in the area of combinatorial game theory. The solution to the ordinary game of Nim has been known for many years and lends itself to numerous other solutions to combinatorial…
We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a set of primes P) as generic structures, arising from a natural game in which two players alternate in building an inverse sequence of surjections. The second player…
Given $n$ piles of tokens and a positive integer $k \leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and…
We consider the strong Ramsey-type game $\mathcal{R}^{(k)}(\mathcal{H}, \aleph_0)$, played on the edge set of the infinite complete $k$-uniform hypergraph $K^k_{\mathbb{N}}$. Two players, called FP (the first player) and SP (the second…
In this paper we are interested in the motion of a ball inside a billiard table bounded by a particular smooth curve. This table belongs to a family of billiards which can all be drawn by a common process: the so-called gardener's string…
A Hist in a cubic graph $G$ is a spanning tree $T$ which has only vertices of degree three and one. A snark with a Hist is called a Hist-snark, see \cite{HO}. We present several computer generated Hist-snarks which form generalizations of…
We discuss the game of Zombie Dice, published by Steve Jackson Games. This game includes green, yellow, and red dice. Each die has brain, footprint, and shotgun symbols on it, with each color of dice having a different amount of each…
We prove for every $n\ge4$ the existence of an $n$-player game in normal form with integer payoffs that has a unique Nash equilibrium, which is fully mixed. In the equilibrium, each probability weight is an algebraic number of degree…
The classical game of {\sc Nim} can be naturally extended and played on an arbitrary hypergraph $\cH \subseteq 2^V \setminus \{\emptyset\}$ whose vertices $V = \{1, \ldots, n\}$ correspond to piles of stones. By one move a player chooses an…
A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum _{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$, $\ldots$,…