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

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Lights Out game over GF(p)

On Jaap's Puzzle Page http:// www.jaapsch.net/puzzles/lomath.htm#domtilings Theorem 7 says: If standard Lights Out is played on a m x n grid-like board, ...
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0answers
149 views

Nimbers and Surreal Numbers [closed]

I have been researching Combinatorial Game Theory. One common theme is the assignment of values to games in order to classify the game as a win for a specific player. One such way is class of surreal ...
5
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1answer
229 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
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0answers
99 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
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3answers
653 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
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2answers
237 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 ...
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0answers
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
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1answer
99 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
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1answer
234 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
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1answer
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?
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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 ...
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1answer
391 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 ...
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475 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
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1answer
757 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
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2answers
333 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 ...
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1answer
721 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
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1answer
212 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
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3answers
440 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
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1answer
738 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
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1answer
284 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.
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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.
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1answer
169 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
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1answer
269 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
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1answer
235 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
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1answer
625 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 ...
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1answer
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 ...
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1answer
732 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 ...
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3answers
222 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 ...
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1answer
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 ...
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4answers
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
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1answer
437 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
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2answers
751 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
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7answers
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
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3answers
805 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
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0answers
364 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
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4answers
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 ...
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3answers
675 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 ...
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2answers
438 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 ...
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1answer
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 $\mathbb{Z}[x]/(F)$ ...
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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 $\lbrace T_i: i\in\omega\rbrace$. ...
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1answer
238 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 ...
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0answers
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
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1answer
729 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 ...
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1answer
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 ...
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0answers
588 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
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3answers
717 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
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1answer
378 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 ...
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2answers
621 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
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1answer
639 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 ...
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2answers
560 views

Game on undirected graphs

One of my friends suggested the following 2-player game. ...