Related papers: Mis\`ere Greedy Nim and Mis\`ere Bounded Greedy Ni…
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 determine the mis\`{e}re equivalence classes of Nim positions under two equivalence relations: one based on playing disjunctive sums with other impartial games, and one allowing sums with partizan games. In the impartial context, the…
The study of the combinatorial game Nim and its variants is rich and varied, but little is known of the game Nim with a Pass. It is Nim, but once per game a player is permitted to skip their turn but this can only be done if a nonempty pile…
This paper introduces a variant of the impartial combinatorial game nim, called tree nim, as well as a particular case of tree nim called tripod nim. A certain existence-uniqueness result and a periodicity result are proven about the…
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…
We consider mis\`{e}re Nim as a normal-play game obtained from Nim by removing the terminal position. While explicit formulas are known for the Sprague-Grundy functions of Nim and Welter's game, no explicit formula is known for that of…
Given an impartial combinatorial game G, we create a class of related games (CIS-G) by specifying a finite set of positions in G and forbidding players from moving to those positions (leaving all other game rules unchanged). Such…
We research a combinatorial game based on the Cookie Monster problem called the Cookie Monster game that generalizes the games of Nim and Wythoff. We also propose several combinatorial games that are in between the Cookie Monster game and…
In this paper, we consider a modular extension to the game of Nim, which we call $m$-Modular Nim, and explore its optimal strategy. In $m$-Modular Nim, a player can either make a standard Nim move or remove a multiple of $m$ tokens in…
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…
We study combinatorial games under mis\`ere convention. Several sets of games have been considered earlier to better understand the behaviour of mis\`ere games. We here connect several of these sets. In particular, we prove that comparison…
This article concerns the resolution of impartial combinatorial games, and in particular games that can be split in sums of independent positions. We prove that in order to compute the outcome of a sum of independent positions, it is always…
Candy Nim is a variant of Nim in which both players aim to take the last candy in a game of Nim, with the added simultaneous secondary goal of taking as many candies as possible. We give bounds on the number of candies the first and second…
In this paper we will be examining impartial scoring play games. We first give the basic definitions for what impartial scoring play games are and look at their general structure under the disjunctive sum. We will then examine the game of…
We announce misere-play solutions to several previously-unsolved combinatorial games. The solutions are described in terms of misere quotients--commutative monoids that encode the additive structure of specific misere-play games. We also…
A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to obtain the outcome, which player has a winning strategy, of a given combinatorial game. Positions of…
Combinatorial games are played under two different play conventions: normal play, where the last player to move wins, and \mis play, where the last player to move loses. Combinatorial games are also classified into impartial positions and…
Past efforts to classify impartial three-player combinatorial games (the theories of Li and Straffin) have made various restrictive assumptions about the rationality of one's opponents and the formation and behavior of coalitions. One may…
Nim is a well-known combinatorial game in which two players alternately remove stones from distinct piles. A player who removes the last stone wins under the normal play rule, while a player loses under the mis\`ere play rule. In this…
In this note, we investigate combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random). In this model, we provide closed-form expressions for the expected number of turns in a…