**5**

votes

**1**answer

191 views

### Is it possible to evaluate Connect 4 positions with Combinatorial Game Theory?

The surreal numbers in Combinatorial Game Theory only work for certain classes of games (e.g. they must satisfy normal play convention). This rules out even reasonable games with fairly ...

**4**

votes

**0**answers

90 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 ...

**8**

votes

**3**answers

633 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 ...

**6**

votes

**2**answers

235 views

### A (possibly boring) Voronoi Game

The board for this game is a compact convex region $\cal C$ of $\mathbb{R}^2$.
Below I illustrate with $\cal C$ an equilateral triangle.
Two players, $A$ and $B$, alternate turns.
At each turn they ...

**4**

votes

**0**answers

91 views

### Analysis of Nim-Like Game? [closed]

There are a finite number of heaps, each with a finite number of counters. Two players take turns; on each move, they may remove exactly one counter from any heap, and also, if the heap is of size ...

**5**

votes

**1**answer

97 views

### Anything known about the Grundy Ordinal of Sylver's Coinage

Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia:
The two players take turns naming positive integers that are not the
...

**8**

votes

**1**answer

230 views

### Yet another Erdős–Szekeres game

Given $n$. Two players in turn write different real numbers $x_1,x_2,x_3,\dots$
The player after whose turn there is a monotone subsequence of length $n$ loses.
I guess that the question 'who wins' ...

**8**

votes

**1**answer

483 views

### Erdős-Szekeres game

Given $n$. Two players in turn mark points on the plane. No three may be collinear, no $n$ may form a convex $n$-gon. The player who does not have legal move loses. Who has a winning strategy?

**2**

votes

**0**answers

85 views

### On subset of Deterministic games

Denote strings $u,v$ from $\{0,1\}^n$.
Denote concatenated pair $[uv]$.
Denote
$$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$
collection of pairs with Hamming distance $1$ from $[uv]$ string ...

**1**

vote

**1**answer

388 views

### Is it possible to create an infinite sequence in which no subsequence is repeated 3 times in a row?

In Chess, there is the Threefold Repetition rule where if a sequence of moves is repeated 3 times in a row, either player can claim a draw.
Say two players wanted to play a legal, infinite game of ...

**14**

votes

**0**answers

469 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
...

**16**

votes

**1**answer

756 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 ...

**10**

votes

**2**answers

326 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 ...

**-1**

votes

**1**answer

711 views

### Maximal score for the 2048 game [duplicate]

t's been weeks (months?) since the 2048 game--by Gabriele Cirulli--took Internet by storm. I have an explicit integer $X$ which is greater or equal than any score of this game. Possibly my $X$ is the ...

**5**

votes

**1**answer

211 views

### Duration and critical groups order in sandpile models and chip firing games

The famous chip firing game (which is closely related to sandpile models) goes like this:
Place chips at the vertices of a graph. REPEATEDLY: If a vertex $v$ of
degree $d_{v}$ has at least ...

**7**

votes

**3**answers

431 views

### Decidability of the winning-position problem in an infinity chess with a finite number of short-range pieces only

Definitions
Long-range pieces: queens, rooks, bishops.
Short-range pieces: pawns, knights, kings.
We can extend the definition of short-range pieces to include also fairy pieces like: Berolina ...

**16**

votes

**1**answer

731 views

### Paul Erdős: 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".
...

**4**

votes

**1**answer

280 views

### Nash Equilibrium in general graphical game

Any one has any ideas about how to compute the Nash Equilibrium in general graphical game? Especially, when the graph structure is not a tree.

**3**

votes

**0**answers

1k views

### How many combinations does Android pattern have? [closed]

Rules-
1) At-least 4 and at-max 9 dots must be connected.
2) There can be no jumps
3) Once a dot is crossed, you can jump over it.

**1**

vote

**1**answer

166 views

### The original proof of Wythoff's game

I am looking for the original proof of Wythoff's game. Wythoff provided the first full analysis of this game in "A modication of the game of nim, Nieuw Archief door Wiskunde, 199{202, 1907". However, ...

**2**

votes

**1**answer

268 views

### efficient arithmetic with (short) Conway games?

We consider "games" in the sense of ONAG. Conway's definition of a game $G$ as a pair $G = \{L \mid R \}$ of sets of games, together with the definitions of inequality and the arithmetic operations ...

**7**

votes

**1**answer

234 views

### Infinite-dimensional hex

Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they ...

**17**

votes

**1**answer

620 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 ...

**0**

votes

**1**answer

172 views

### What is a description of winning strategies in this tile game?

I'm hoping someone can help me figure out how to describe all winning strategies for "Player 1" in the following game:
Consider a board with $n$ tiles arranged in a row. Player 1 and Player 2 each ...

**0**

votes

**1**answer

725 views

### Calculate the probability of winning for a selected tic-tac-toe player

I am not a mathematician, I am a programmer. Sorry, if formulation of the problem is inexact.
I want to calculate the probability of winning for a selected tic-tac-toe player.
I have a directed graph ...

**2**

votes

**3**answers

221 views

### Simulating Mixed Nash Equilibria

I have a $N$ person game where each person has a set of $M$ discrete strategies. I know from the theory that at least one mixed strategy Nash Equilibrium exists.
Can someone please tell me how do I ...

**1**

vote

**1**answer

213 views

### Infinite board games: sentences about

As a unified approach if we have an ( read any) infinite board game described as $\mathcal{G}$ using a particular axiom set A..
can a sentence be devised in A which automatically answers the basic ...

**1**

vote

**4**answers

3k views

### How many possible ways are there to win in Quoridor?

Quoridor is a board game in which the objective is to move a piece across to the other side. A player can put up fences to block other players from advancing forward. How many possible ways are there ...

**9**

votes

**1**answer

435 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 ...

**23**

votes

**2**answers

746 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 ...

**38**

votes

**7**answers

3k 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 ...

**8**

votes

**3**answers

787 views

### Mathematical model for Hanoi Towers

The strategy for the Hanoi Tower puzzle is quite simple. It is based on parity only. In an $n$-pieces puzzle, $2^n-1$ moves are sufficient to carry the whole pile from one pole to another one. My ...

**5**

votes

**0**answers

362 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 ...

**30**

votes

**4**answers

3k 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 ...

**3**

votes

**3**answers

655 views

### Motivation and Intuition for Sprague-Grundy Theorem

I have read about Sprague Grundy Theorem and understand the proof of its correctness. However, I am unable to see the motivation behind the definitions. How do Sprague and Grundy know that they should ...

**0**

votes

**2**answers

431 views

### Generalized Sprague-Grundy Theorem

Hey,
I know what is Sprague-Grundy theorem, but I want to know about generalized Sprague-Grundy (GSG) theorem ( which is used for games with cycles ). Apparently there seems to be very less ...

**14**

votes

**1**answer

796 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 $\mathbb{Z}[x]/(F)$ ...

**2**

votes

**0**answers

112 views

### Open games formed by pasting together infinitely many clopen games

Throughout, I think of games and their underlying trees as the same: so a "clopen game" and a "well-founded tree" mean the same thing.
Fix a sequence of clopen games $\lbrace T_i: i\in\omega\rbrace$. ...

**0**

votes

**1**answer

236 views

### Equilibrium of random zero-sum game,

Hi,
How to find, or at least express, the equilibrium of a zero-sum game with an $n*n$ payoff matrix (each player has $n$ strategies) and the payoff of the entry $(i,j)$ is $u(i,j)$. $u$ a random ...

**1**

vote

**0**answers

119 views

### What is known about infinite diminished disjunctive compounds of loopfree partizan combinatorial games?

Background
Basic theories of loopy (normal-play) games which may go on forever under the usual disjunctive sum (the game ends when there are no moves available for you in any component on your turn) ...

**21**

votes

**1**answer

728 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 ...

**0**

votes

**1**answer

276 views

### a question on game of chess [closed]

In the game of chess,it is a proven fact that either of the three conditions hold:
1)white has a winning strategy
2) black has a winning strategy or
3)either of them can at least force a draw.
It is ...

**0**

votes

**0**answers

555 views

### Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...

**3**

votes

**3**answers

714 views

### Generalized tic-tac-toe

We begin with $2n+1$ cards, each with a distinct number from $-n$ to $+n$ on it, face up in between the two players of the game. The players take turns selecting a card and keeping it. The first ...

**5**

votes

**1**answer

376 views

### Resources-Aware Combinatorial Game Theory

First of all, I preemptively apologize if my question happens to be naive, I am no expert of CGT (or general game theory, for that matter).
Now the question:
**is there such a thing as the study of ...

**3**

votes

**2**answers

611 views

### The game of removing two vertices in a graph

Consider the following impartial combinatorial game played with finite graphs: A move removes two adjacent vertices; and of course all edges connected with them. The game then continues with the new ...

**10**

votes

**1**answer

638 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 ...

**2**

votes

**2**answers

536 views

**2**

votes

**3**answers

1k views

### Motivation for the Sprague-Grundy theorem

The Sprague-Grundy theorem states that every impartial combinatorial game under the normal play convention is equivalent to a (unique) nimber.
What does the equivalence relation thus defined tell us ...

**3**

votes

**3**answers

540 views

### Chomp! without the law of the excluded middle

Consider the following impartial combinatorial game, a generalization of Chomp! as mentioned in the paper by David Gale: At each step, we have a finite partially ordered set $S$. Player I or II ...