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.

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Hex game winning strategy

I was teaching myself how to play a hex board game by reading some books a couple days ago. I learned how to do $2$ x $2$ and $3$ x $3$ hex games by starting at the principal diagonal. I wanted to ...
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Optimal Tic Tac Toe algorithm without lookahead

Is there any algorithm for tic tac toe that does not rely on a lookahead algorithm that is perfect for any sized boards? Edit: For boards larger than $3 \times 3$, we have to find the best move for ...
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A game problem- double or increment by 1

Its a two player game. Initially $P=1$, and there is some fixed integer $Q>1$. A valid move consists of either increasing $P$ by $1$ or doubling it iff on doing so $P$ does NOT exceed $Q$.The ...
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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 ...
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Reverse Hex board game winning strategy

I just wanted to know the winning strategy to this question: In a reverse Hex board game I know it means where the player who first forms a path between his/her edges loses. Find a winning ...
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116 views

Play with pairs of numbers

Two players are playing a game. The game is played on a sequence of positive integer pairs. The players make their moves alternatively. During his move the player chooses a pair and decreases the ...
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On a game with cuisenaire rods

Here is a variation of a Nim game : consider a full set of cuisenaire rods - 10 rods of all integer lengths between 1 and 10. Set a number N between 1 and 54. Player 1 choose one of the 10 rods and ...
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Subtraction Game

I recently read about the Nim Subtraction Game. I have a variant, Suppose you have N stones and two players Alice and Bob, who can choose to pick either 1 stones or K stones. If Alice plays first when ...
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Optimal strategy in a match picking game

I faced this exercise making a practice exam and couldn't find the answer: Question: 2 players play the following game with matches. Each player can take in turn some matches from a bunch of ...
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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 ...
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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 ...
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“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
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Saddle Points on Matrices

Let $n$, $m$ be positive integers. Suppose that $A$ is a $2$ x $n$ or an $m$ x $2$ matrix and that it has a saddle point. Show that among the saddle points of $A$ there exists at least one which ...
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Nimber of selective compound games

Background/Definitions. Let $\alpha,\beta$ ordinal numbers. The Hessenberg sum $\alpha \# \beta$ is defined recursively as the smallest ordinal which is $>\alpha' \# \beta$ and $> \alpha \# ...
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Chat Noir solvable?

There is a relatively simple flash game that I enjoy playing -- http://www.gamedesign.jp/flash/chatnoir/chatnoir.html is one version of it, though I've found many -- and it centers around trying to ...
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Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
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Efficiently count possible nim-like moves

Consider $n$ piles of coins, with pile $i$ having $a_i$ coins. A valid move is to remove zero or more coins from each of the piles, with the constraint that atleast one pile should remain unchanged, ...
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Number Theoretic Game

2 players A and B play a game. At the start of the game, $n$ positive integers (not necessarily distinct) are written on a notebook. First, player A chooses a number from the notebook and declares it ...
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The point-open game and $\omega$-covers

Let $X$ be a topological space. The point-open game $G_{po}(X)$ is defined as folows. It is played by two players ONE and TWO. In the n'th step $(n \in \omega)$, ONE choose a finite subset $F$ of ...
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News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
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How many different Forsyth-Edwards Notation ranks are possible?

(This is a combinatorics question, and therefore more appropriate here than at the Chess Stack Exchange.) Background: in chess, board positions are recorded using a system called Forsyth-Edwards ...
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Terminology questions about a game where one may “save his progress” at the cost of a turn.

The game is for $p$ players who each start at square $1$. Each turn, a player can either roll an $m$-sided dice or place a marker on his current square. If he rolls $x\in\{2,\ldots, m\}$, he ...
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How many different chess-board situations can occur?

If you play a standard chess game on a normal $8 \cdot 8$ chess board with the usual rules: How many different "board representations" can exist? Upper bound: Well, you have 16+16 = 32 chess pieces ...
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Computation of a 3-dimensional game matrix

For natural numbers $n_1 \leq n_2 \leq n_3$ we define $\beta(n_1,n_2,n_3)$ recursively to be the smallest natural number which is not among the numbers $\beta(m_1,m_2,m_3)$, where $m_1 \leq n_1 \leq ...
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What is the highest possible score in 2048 hard?

There is a variant of the popular game 2048, called 2048 hard or 2048 impossible, which automatically places each new tile in the hardest possible location. Is this variation possible to solve, and if ...
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What is the optimal strategy for this 2 player game?

Let some finite array of integers is given initially. There is a number k which is initially '0'. In a move, a player will select a number from the array arr[i] and change k to gcd(k,arr[i]). Also, ...
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The name of a game from the 2013 Putnam

Does the following game from the 2013 Putnam have an official name? Based on my searches, it seems to have been created just for the exam. Let $n\geq 1$ be an odd integer. Alice and Bob play the ...
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Stable Marriage Problem

I would just like clarification for the following problem: Suppose $M_1$ and $M_2$ are two stable matchings between n men and n women, and we allow each woman to choose between the man she is paired ...
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What are the chances of shooting the moon in Hearts?

I'm posting this again here from boardgames.SE because it was suggested there that this is the more appropriate question to ask it. So, that said, here's my question: given a fair shuffle, that all ...
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Unavoidable structure of this kind of function:$Z\rightarrow N$.

Suppose that $f$:$Z\rightarrow N$ is a surjection and $|f^{-1}(n)|=2$ for every $n\in N$. I found that there is $n\in Z$ such that $f(n)$, $f(n+1)$, $f(n+4)$, and $f(n+5)$ differ from each other. ...
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Number of ways to make first move

Alice and Bob are playing a game. They have N containers each having one or more chocolates. Containers are numbered from 1 to N, where ith container has A[i] number of chocolates. The game goes like ...
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What is the optimal strategy?

There are $m+n+1$ cards numbered $1,2,\ldots m+n+1$. Participants A and B respectively get $m$ and $n$ cards. Meanwhile, they only know what they get. The remaining card is face down on the desk. ...
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How many lines needed to not lose in tetris game?

Suppose we play a tetris game with tetris be given randomly. Is there exists a number of lines that we can play infinitely, i.e. do not lose the game?
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Does any $\omega$-cover in which $X \in L(\mathcal U)$ is also a $\gamma$-cover?

As a continuation to this question: An $\omega$-cover, is an open cover $\mathcal U$ of $X$, such that, $X \notin \mathcal U$, and for every finite set $F \subset X$, there exists an open set $U ...
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Proof that 12 in a row tic-tac-toe is a tie game?

How can be it proved that tic-tac-toe on an infinite grid (winning with 12 in a row, a column or a diagonal) can always end in a tie (with optimal strategies of both players)? There is a hint: to use ...
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Any known strategies for toads and frogs?

Are there any heuristic strategic for playing Toads and Frogs known? I reckon the optimal playthrough may be hard to achieve due to the game being NP-hard but at least something that regularly ...
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What are the available libraries or programs for finding extremes of a function with no symbolic definition?

In my current mathematical inquiry, I would like to gain insight on behaviour of a $d$-dimensional continuous function by locating its maximum over the hyperplane $\sum_{i=1}^d x_i = 1$ for $x_i$ ...
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More than Nim (combinatorics problem)

A two-player game is played with two piles of stones, with sizes m,n. On a player's turn, that player can remove any positive number of stones from one pile, or the same positive number of stones from ...
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S-Grundy Number

Help needed in understanding S-Grundy Number , any good tutorial. I am trying to solve Mathalon Problem 146 S-Grundy Game
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Matrix Saddle Points and Dominance

I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it can be used in games. I wanted to know how to prove a problem like ...
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Proof of Conway's “Simplicity Rule” for Surreal Numbers

A "number" in the sense of Combinatorial Game Theory is a game $G = \{ a,b,c,\dots | \; d,e,f,\dots \}$ such that $a,b,c < d,e,f$. Then our game is between the left and right options: $$ a,b,c ...
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de Bono's L-game modification

I am trying to find out if a simple modification od de Bono's L-game is still infinite if two players are perfect. Modified rule is that there no neutral pieces but, there is one piece for each player ...
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Interesting Probability Game With Uneven Return Scenarions

Let say we play a game. The game which spans the course of 5 trials. The game is as follows. You either receive 100 points or 40 points as a final payout. The only time a payout of a 100 points occurs ...
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Recursive core of coalition game

Can someone please explain the recursive core concept possibly with an example? http://arno.unimaas.nl/show.cgi?fid=5152 I don't understand how the recursion works. Thank you
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What algorithms do you know for beltway reconstruction?

I've faced the beltway reconstruction problem and I've developed a simple backtrack algorithm, what algorithms do you know for this problem? Beltway Reconstruction Problem: Assume there is a set of ...
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Mathematical reason for 2-player turn-based games

I've been reading Games, Puzzles, and Computation which analyzes games through game theory and complexity theory. The authors introduce something called "Constraint Logic" as a way of modeling games ...
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Applications of Scoring Play Combinatorial Game Theory

I'm currently looking into economic applications of scoring play combinatorial game theory. Details of the theory can be found in this paper. http://arxiv.org/abs/1202.4653 A friend of mine ...
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Can “proving mathematical theorems” as a 1 player puzzle be studied in combinatorial game theory?

In Combinatorial game theory other than 2 player games 1 player puzzle games are studied too. proving a mathematical theorem is a 1 player puzzle game. So I was curious to know whether it can be ...
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Nimnumber of three pile Wythoff Game

Wythoff Game is where there are n-piles of stones and two players. Players take turn removing m stones from one pile, or m stones from more than one pile. The player who picks up the last stone ...