# Questions tagged [combinatorial-game-theory]

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

**190**

votes

**3**answers

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

**11**answers

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

**5**answers

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

**4**answers

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

**2**answers

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

**4**answers

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

**7**answers

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

**3**answers

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

**2**answers

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

**5**answers

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

**1**answer

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

**1**answer

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**

votes

**4**answers

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

**2**answers

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

**1**answer

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

**3**answers

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

**6**answers

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

**1**answer

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

**4**answers

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**

votes

**1**answer

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

**3**answers

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**

votes

**4**answers

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

**1**answer

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

**2**answers

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**

votes

**1**answer

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

**4**answers

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

**3**answers

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

**5**answers

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

**5**answers

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

**3**answers

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

**1**answer

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**

votes

**0**answers

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

**5**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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**

votes

**12**answers

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

**1**answer

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

**1**answer

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

**4**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**2**answers

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 ...