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Questions tagged [combinatorial-game-theory]

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

190
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3answers
11k views

A Game on Noetherian Rings

A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring $\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ...
91
votes
11answers
12k views

Checkmate in $\omega$ moves?

Is there a chess position with a finite number of pieces on the infinite chess board $\mathbb{Z}^2$ such that White to move has a forced win, but Black can stave off mate for at least $n$ moves for ...
54
votes
5answers
7k views

Decidability of chess on an infinite board

The recent question Do there exist chess positions that require exponentially many moves to reach? of Tim Chow reminds me of a problem I have been interested in. Is chess with finitely many men on an ...
47
votes
4answers
8k views

Do there exist chess positions that require exponentially many moves to reach?

By "chess" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an $n\...
43
votes
2answers
4k views

Does knight behave like a king in his infinite odyssey?

The Knight's Tour is a well-known mathematical chess problem. There is an extensive amount of research concerning this question in two/higher dimensional finite boards. Here, I would like to tackle ...
43
votes
4answers
6k views

Verifying the correctness of a Sudoku solution

A Sudoku is solved correctly, if all columns, all rows and all 9 subsquares are filled with the numbers 1 to 9 without repetition. Hence, in order to verify if a (correct) solution is correct, one has ...
43
votes
7answers
6k views

Conway's game of life for random initial position

What is the behavior of Conway's game of life when the initial position is random? -- We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus). Specifically ...
37
votes
3answers
4k views

A game on integers

$A$ and $B$ take turns to pick integers: $A$ picks one integer and then $B$ picks $k > 1$ integers ($k$ being fixed). A player cannot pick a number that his opponent has picked. If $A$ has $5$ ...
33
votes
2answers
2k views

Who wins two player sudoku?

Let's say players take turns placing numbers 1-9 on a sudoku board. They must not create an invalid position (meaning that you can not have the same number in within a row, column, or box region). The ...
30
votes
5answers
2k views

Can one make high-level proofs about chess positions?

I realize this question is risky (as the title and the tags indicate), but hopefully I can make it acceptable. If not, and the question cannot be salvaged, I'm sorry and ready to delete it or accept ...
30
votes
1answer
2k views

Is there a position in infinite Go for which the life of a particular stone has transfinite game value?

As follow up to Checkmate in $\omega$ moves?, we can ask the same question about go. Is there a position on a $\mathbb Z \times \mathbb Z$ goban such that either black can kill a white group, but ...
28
votes
1answer
776 views

Vanishing line on Conway's game of life

If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$. ...
27
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4answers
3k views

Alice and Bob playing on a circle

I want to solve this problem: Let there be $n \ge 2$ points around a circle. Alice and Bob play a game on the circle. They take moves in turn with Alice beginning. At each move: Alice takes ...
25
votes
2answers
1k views

Is there any superstable configuration in the game of life?

This question spins off of Gil Kalai's recent question on Conway's game of life for a random initial configuration. There are numerous configurations in the game of life that are known to be stable-...
25
votes
1answer
2k views

Who wins this two-player game based on the sandpile model?

Given a connected graph $G$, two players, Blue and Green, play the following game: initially, all vertices are unclaimed. Players alternate turns. On her turn, Blue adds a token to either an ...
23
votes
3answers
2k views

Rolling-ball game

The analyses in two recent MO questions ("recent" with respect to the original posting in 2011), "Rolling a random walk on a sphere" and "Maneuvering with limited moves on $S^2$," suggest a Rolling-...
22
votes
6answers
3k views

Neutral tic tac toe

I heard this puzzle from Bob Koca. Suppose we play misere tic-tac-toe (a.k.a. noughts and crosses) where both players are X. Who wins? That particular puzzle is easy to solve, but more generally, ...
22
votes
1answer
973 views

Sane bound on number of moves for Maker-Breaker game on $\mathbb R^2$ for $\{0,1,2,3,4\}$

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (...
21
votes
4answers
2k views

The 1-step vanishing polyplets on Conway's game of life

A $n$-polyplet is a collection of $n$ cells on a grid which are orthogonally or diagonally connected. The number of $n$-polyplets is given by the OEIS sequence A030222: $1, 2, 5, 22, 94, 524, 3031, \...
20
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1answer
1k views

A Ramsey avoidance game

Consider the following game: Given $K_n$ the complete graph on $n$ vertices, two players take turns coloring its edges. Initially no edges are colored. At his turn a player can color a prevoiusly not ...
19
votes
3answers
510 views

The arithmetic progression game and its variations: can you find optimal play?

Consider the arithmetic progression game, a two-player game of perfect information, in which the players take turns playing natural numbers, or finite sets of natural numbers, all distinct, and the ...
18
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4answers
4k views

David Gale's subset take-away game

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: ...
18
votes
1answer
949 views

Removing pawns - the game

Here is a simple game I've invented (if the idea is not fresh, then please let me know): The game is played on a board. The board has some (finite) number of lines drawn on it. A pawn is placed on ...
18
votes
2answers
1k views

Determine or estimate the number of maximal triangle-free graphs on $n$ vertices

Among the collections of the open problems of Paul Erdős on the website of Professor Fan Chung, there is one called "number of triangle-free graphs". http://www.math.ucsd.edu/~erdosproblems/erdos/...
18
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1answer
441 views

Who wins the Rubik's cube game?

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube (to make the problem easier). Solver and Spoiler take turns making 90 degree twists (starting with Solver). ...
17
votes
4answers
2k views

Variation on a matrix game

The original problem appeared on last year's Putnam exam: "Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays first. At each turn, ...
17
votes
3answers
1k views

Traversing the infinite square grid

Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move $n$, one must jump $a_n$ steps in one of the directions north,south,east or west, and mark ...
17
votes
5answers
869 views

When is a game tree the game tree of a board game?

This question arises from what I find interesting in the recently asked question What is a chess piece mathematically? My answer to that question was that mathematically, game pieces are in general ...
16
votes
5answers
3k views

Irreversible chess

Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ...
16
votes
3answers
1k views

Why does the bitxor function appear in Nim?

I am conducting research in Combinatorial Game Theory (CGT). Although I have done a considerable amount of reading, I do not completely understand why the bit-xor function also known as the nim-sum ...
16
votes
1answer
1k views

Playing an (invertible) matrix game with two players

Players $A$ and $B$ take an empty $n \times n$ matrix and place, one by one, an element (say, a rational number) in an unoccupied place of this matrix. Player $A$ starts. The game ends if there is no ...
16
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0answers
830 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 ...
15
votes
5answers
2k views

Nimber multiplication

Is there a game-theoretic interpretation of nimber multiplication? There is such for addition (a single move in a+b is either a move in a or a move in b).
15
votes
1answer
1k views

Principal maximal ideals in Z[x]/(F)

Is there some irreducible $F \in \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/(F)$ has no principal maximal ideal? Equivalently, is it possible that the $1$-dimensional integral domain $\mathbb{Z}[x]/(F)$ ...
15
votes
1answer
442 views

Is the game Hanabi NEXPTIME-complete?

The game Hanabi is a cooperative, hidden-information game. You can read the rules elsewhere, but broadly speaking the players are attempting to cooperatively build a fireworks display by playing cards ...
14
votes
1answer
1k views

Mathematical solution for a two-player single-suit trick taking game?

The question on games and mathematics that appeared recently on mathoverflow (Which popular games are the most mathematical?) reminded me of a problem I encountered some time ago : starting with the ...
14
votes
2answers
13k views

Winning strategy at chomp (a chocolate bar game)?

The game of chomp is an example of a game with very simple rules, but no known winning strategy in general. I copy the rules from Ivars Peterson's page: Chomp starts with a rectangular array of ...
14
votes
0answers
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 ...
13
votes
1answer
625 views

Bipartite Nim-Geography

Two players are playing a game on a bipartite graph where all of the edges are nim-heaps of various sizes. A token starts on one of the vertices, and on your turn you must move the token over an edge ...
13
votes
0answers
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 ...
12
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12answers
7k views

Are there any interesting connections between Game Theory and Algebraic Topology?

I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? ...
12
votes
1answer
712 views

Sliding blocks puzzle

Consider a 'game' played on a subset $S$ of an $n^2$ square grid as follows. There are 3 types of pieces, each occupying a square of $S$, 1 green, some red and the rest are blue, a move consists of ...
11
votes
1answer
841 views

The infinite X in Conway's game of life

In Conway's game of life, take the initial position to be two infinite diagonal lines of live cells, with a single cell in common. Does this thing converge to a stable configuration? I.e., is the ...
11
votes
4answers
912 views

a game on numbers

Hello, here is a little two-players game. Players A and B choose three numbers : a, b and c for A, a', b' and c' for B. The values are numbers between 0 and 1, their sum is 1, and they are ordered: $...
11
votes
2answers
1k views

Can anyone analyze this misere game?

Problem Let $* = \{0\}$ be the one matchstick nim game, let $*2 = \{0,*\}$ be the two matchstick nim game, let $*3 = \{0,*,*2\} = *2+*$ be the three matchstick nim game, let $g = \{0, *2+*3, *2+*2+*2\...
11
votes
1answer
178 views

A game: building a bounded degree graph

Given positive integers $n$ and $d\leqslant n-1$. Two players build a graph, starting with $n$ vertices and no edges. On each turn, a player joins two yet not joined vertices by an edge. It is ...
10
votes
1answer
759 views

What does “game theory” cover and how should it be called?

There seems to be a huge discrepancy in what people refer to when they speak of "game theory". I tend to think of it as including, among other things: Combinatorial game theory dealing with certain ...
10
votes
2answers
217 views

For which number of pairs is it an advantage to start in memory

Players A and B play memory starting with $n$ pairs of cards. We assume that they can remember all cards which have been turned. At his turn a player will first recall if two cards already turned ...
10
votes
2answers
483 views

Nim and the Sierpinski Gasket

(I discovered this in high school, sent it off to a journal, never heard back, and forgot about it. I've never found anyone else who appeared to know about it; the combinatorial game theorists I've ...
10
votes
2answers
455 views

What surreal numbers are representable by Red-Blue Hackenbush games?

Every game of Red-Blue Hackenbush represents a surreal number. Is the converse true? Assuming that it is false, what can be said about the class of surreal numbers that are representable by such ...