# Questions tagged [combinatorial-game-theory]

Two-player turn-based perfect-information games, surreal numbers, impartial games and Sprague-Grundy theory, partizan games

37 questions
831 views

### Are the moves/rules of standard chess delicately balanced?

(While the world chess championship is in progress in Sochi...) Is there mathematical evidence that standard chess is somehow ...
709 views

### A Combinatorial Game: the Snake and the Hunter

The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both ...
1k views

### Characterizing the surcomplex numbers

Conway showed that the Field of surreal numbers ("${\bf No}$") is the maximal totally ordered Field. Later Jacob Lurie showed that the Group of all partizan games ${\bf Pg}$ is the universally ...
156 views

Let $G = (V, E)$ be a finite graph, and $S \subseteq V$ initially be an empty set. Alice and Bob play a game, making moves in turns starting with Alice. A move consists of choosing a vertex $v \in V \... 0answers 146 views ### Is there a better way to understand this particle? I've been reading through Winning Ways and was working through some examples of my own related to cooling and particles, and I managed to stump myself. If we let ... 0answers 449 views ### A Banach-Tarski game This is partially inspired by the question https://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written. A paradoxical family of subsets ... 0answers 1k views ### Is there a chess position equivalent to the Collatz conjecture? Suppose we have an infinite board with a finite number of chess pieces. The question is whether white can checkmate black (without the after 50 moves it is a draw rule). Can you give an explicit ... 0answers 82 views ### Length of optimal play in Hex as a function of size Consider Hex on an$n \times n$board without a swap rule, so that the first player wins. Assume the first player tries to minimize the length of the game, and the second player tries to maximize the ... 0answers 129 views ### Combinatorial game similar to Sprouts Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy? Basically this game is "Sprouts without midpoints". One starts with$n$points in the ... 0answers 473 views ### Number of Configurations in the optimal Hanoi tower There is a unique strategy how to move$n$disks from the first rod to the second optimally and it takes$2^n-1$steps, solution is obtained by simple recursion. I am interested into the following ... 0answers 128 views ### Games in non-standard models Has anyone studied Combinatorial game theory in non-standard models? In particular, we can work in either non-standard models of set theory, or we can work in non-standard models of arithmetic, where ... 0answers 187 views ### A winning move for the first player in$3 \times 3 \times \omega$Ordinal Chomp I have been trying to analyse the game of Ordinal Chomp played on a$3 \times 3 \times \omega$board. The rules can be found in the Wikipedia article, briefly: This game is played between two players ... 0answers 209 views ### Topological Subset Take-Away David Gale's subset take-away game is a game where two players A and B play with a finite set$S$. Players alternately choose proper nonempty subsets of$S$such that if a subset is chosen, then none ... 0answers 198 views ### Generalization of Sprague-Grundy Theorem In my research on Combinatorial Game Theory, I used a certain theorem that is essentially a generalization of the Sprague-Grundy theorem. Because the result hinges too much on the work of others to be ... 0answers 202 views ### Mistake in ONAG? In the second edition of the book "On Numbers and Games" by Conway there is a theorem 88 (p. 194) on comparison of sums of${\uparrow} x$games. It contains a weird statement: ... (If$X$is a sum ... 0answers 119 views ### Modern advances in combinatorial game theory I'm going to take part in teaching a course in combinatorial game theory in the best of ONAG's spirit. I was wondering if there are interesting post-ONAG results that are worth mentioning in (a later ... 0answers 247 views ### A game played on binary matrices ($2$-dimension coin-turning game) Let$r\geq 1$be a natural number. I am interested in the following (two-player, impartial, perfect-information) game: The state of the game is an$n\times n$matrix with coefficients in$\mathbb{F}...
This question originates from the paper On the Axiom of Determinateness by Jan Mycielski, section 7. Given a set $X$ and an ordinal $\alpha$, the author defines a transfinite game of length $\alpha$ ...