Combinatorial game theory (abbreviated CGT) is the subfield of combinatorics (not traditional game theory) which deals with games of perfect information such as Nim and Go. It includes topics such as the Sprague-Grundy theorem and is tangentially related to the Surreal Numbers.

# Questions tagged [combinatorial-game-theory]

917 questions

**394**

votes

**1**answer

### The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative Noetherian ring $R$. Player one mods out a nonzero…

Alex Becker

- 58,347
- 7
- 124
- 180

**84**

votes

**6**answers

### Alice and Bob play the determinant game

Alice and Bob play the following game with an $n \times n$ matrix, where $n$ is odd. Alice fills in one of the entries of the matrix with a real number, then Bob, then Alice and so forth until the entire matrix is filled. At the end, the…

pad

- 2,827
- 1
- 17
- 35

**67**

votes

**8**answers

### Why do people lose in chess?

Zermelo's Theorem, when applied to chess, states:
"either white can force a win, or black can force a win, or both sides can force at least a draw [1]"
I do not get this. How can it be proven?
And why do people lose in chess then, if they can…

Brika

- 1,099
- 1
- 8
- 12

**52**

votes

**3**answers

### Sharing a pepperoni pizza with your worst enemy

You are about to eat a pepperoni pizza, which is sliced into eight pieces. Each pepperoni will unambiguously belong to some slice (no pepperoni is "between" slices).
The caveat is that you have to share the pizza with your worst enemy, and you want…

Mankind

- 12,812
- 7
- 30
- 51

**50**

votes

**3**answers

### Prime number construction game

This is a variant of Prime number building game.
Player $A$ begins by choosing a single-digit prime number. Player $B$ then appends any digit to that number such that the result is still prime, and players alternate in this fashion until one player…

scip

- 949
- 10
- 15

**48**

votes

**5**answers

### The expected outcome of a random game of chess?

Imagine a game of chess where both players generate a list of legal moves and pick one uniformly at random.
Q: What is the expected outcome for white?
1 point for black checkmated, 0.5 for a draw, 0 for white checkmated. So the expected outcome…

Rebecca J. Stones

- 24,954
- 2
- 41
- 103

**38**

votes

**9**answers

### Maximum board position in 2048 game

A game called 2048 is making rounds on social media. I am trying to determine the maximum score attainable for this game. Let's assume WLOG that only 2s are returned (if 4s are possible the max score is doubled).
First, I'll relax some of the rules…

Isaac

- 531
- 1
- 4
- 9

**35**

votes

**3**answers

### Three against the devil: a combinatorial game

A team of three sinners plays a game against the devil. They confer on strategy beforehand; then they go into three separate rooms, and there is no more communication between them. The play in each room follows the same rules, as described below.
A…

user75900

**34**

votes

**3**answers

### Guaranteed Checkmate with Rooks in High-Dimensional Chess

Given an infinite (in all directions), $n$-dimensional chess board $\mathbb Z^n$, and a black king. What is the minimum number of white rooks necessary that can guarantee a checkmate in a finite number of moves?
To avoid trivial exceptions, assume…

TROLLHUNTER

- 8,398
- 3
- 41
- 91

**33**

votes

**1**answer

### A 20+ year old combinatorial problem - the cookie game

Learned about this not too long after the time of the original problem publication through a classmate who visited MIT one summer.
http://faculty.uml.edu/jpropp/cookie2.pdf
The problem goes as follows:
Given a set of cookies with finite…

Meina222

- 579
- 3
- 11

**29**

votes

**3**answers

### Hat 'trick': Can one of them guess right?

There are $n$ boys and $n$ girls. Each of them is given a hat of only 4 possible (known) colors and doesn't know its color. Now each can only see all the colors of hats of those of the other gender and no contact is allowed, then each is asked to…

Juggler

- 1,305
- 5
- 18

**28**

votes

**1**answer

### Analysis of a combinatorial game with prime numbers

Many years ago, a coworker showed me a programming problem involving a combinatorial game with prime numbers that he had gotten somewhere or other. (For some reason, he refused to tell me the source.) Actually it is an infinite family of games,…

saulspatz

- 51,472
- 6
- 32
- 63

**26**

votes

**1**answer

### What is the optimal strategy in the "Factor Game"?

Edit (Nov 1, 2015): Bounty awarded, but the full question (i.e., what is the optimal strategy) remains open at the time of this update.
Consider the Factor Game played as follows:
Given a list of positive integers $1, \ldots, n$, two players (red…

Benjamin Dickman

- 13,259
- 2
- 39
- 82

**23**

votes

**3**answers

### A game on a graph

Alice and Bob play a game on a complete graph ${G}$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m,\: 1\leq m \leq…

shadow10

- 5,301
- 13
- 39

**22**

votes

**0**answers

### Analyzing a class of vertex-deletion games

As part of the discussion on this question (Permutation Game Redux), a simple vertex-deletion game was proposed. The game is very simple.
Disconnect. Players alternately remove vertices from a graph $G$. The player that produces a fully…

mjqxxxx

- 37,360
- 2
- 50
- 99