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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Can someone please help me understand what a “player set” is in extensive form game

my text defines player set as: In N-player game $g$, each non-terminating node is partitioned into $N+1$ sets $g^0, ... g^N$. These are player sets. However it makes no attempt to identify ...
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Is this general variant of nim NP-hard to decide who has a winning strategy?

Suppose there are $n$ piles of stones, where pile $i$ originally has $m_i$ stones, and each pile has a maximum number of stones $k_i$ that can be taken on each turn. Fix integer $N \geq 1$. Suppose ...
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39 views

Why does the 1st player in this subset take-away game always have a winning strategy?

This is a HW problem of mine that I cannot, for the life of me, figure out. There is a take-away game where there are a number of elements A, and the person that wins is that last person to remove a ...
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60 views

Proving the rules of a complicated game are well defined

What strategy could one use to formally model a game and prove that the rules do not lead to any self contradiction? A major example that comes to mind is Magic the Gathering. The card ...
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31 views

Sprague–Grundy numbers

I would appreciate if someone can help me understand the columns of the table in this blog: http://lbv-pc.blogspot.co.uk/2012/07/treblecross.html . The author writes he leaves this as an exercise to ...
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47 views

What is the mixed strategy equilibrium bid, if any, for complete information auction games with minimum bid?

Consider the following complete-information, auction game. There are two players $i=1,2$. Each bids simultaneously a value $b_i\in[0,\infty)$. The payoff function is symmetric: $$ \pi_i ...
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18 views

super-additive, sub-additive, and shapely value limitations?

I am working on the coalition formation. Most of the scientist used concept of shapely value for distributing the utility among the members of coalition. Up to my understanding, shapely value is good ...
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550 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
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60 views

NP combination puzzle (Klotski)

I've written a C++ program to solve sliding puzzles games such as UnblockMe and Car Parking. I'm quite happy about it, since it solves various schemes in less than a second. Recently I fed the game ...
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118 views

Maths Puzzle: Partitioning a set into two disjoint sets

Le $X$ be the set of all non-empty subsets of $\{a,b,c,d,e,f\}$. So $X=\{a,b,c,d,e,f,ab,ac,ad,ae,af,bc,bd,be,bf,cd,ce,cf,de,df,ef,abc,\cdots,abcdef\}$; i.e., $|X|=63$. We want to partition $X$ into ...
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Multi-dimensional naughts and crosses: victory for first player?

In a $2$-dimensional checkerboard which is $2\times2$, the player going first necessarily achieves $2$ in a row. In a $3$-dimensional board which is $3\times3\times3$, a player going first using a ...
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51 views

An inequality relating to moves to P-positions in Nim

I have been researching this variant of Nim. I have been unable to prove the following claim. What is annoying is that I feel I am missing something really obvious. Does anyone have any ideas on how ...
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63 views

“On Numbers and Games” or “Winning Ways for Your Mathematical Plays”?

I'm really interested in John Conway's work on games and I want to spend my winter reading something of his but I'm not sure between "On Numbers and Games" or "Winning Ways for Your Mathematical ...
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33 views

How is Wythoff's Theorem proved?

Specifically, how does one prove the following? Suppose $(a,b)$ is not of the form $(A_n,B_n)$, where $A_n=\lfloor n \phi \rfloor$ and $B_n= \lfloor n \phi^2 \rfloor$. Then there is a move in ...
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38 views

Solving a Recurrence for a Mathematical Game

The problem is: Two players take turns removing coins from a pile. There are initially $n$ coins, and on each turn, a player can remove $a_1, a_2, \dotsc, a_k$ coins. The player who cannot remove ...
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163 views

Optimal strategy for 2 players Lights Out game variation

Consider a turn-based game for 2 players. They're both playing on the same board. The board is 8x8, randomly generated and each cell has 0 or 1 (with equal probabilities), for example: ...
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Game on simple finite graphs

Consider the following game on graphs (no multiple edges, but graphs can be disconnected). Players A and B alternate picking a vertex. After picking a vertex, a number is assigned to that vertex such ...
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80 views

How many ways we can arrange numbers such that sum of arrangement will be a given number

we have a value $p$ and we have to arrange numbers at $p$ places such that no number is greater that $p$ and no number is less than $0$ and also sum of arrangement should be a given number (suppose ...
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45 views

A Good book for Combinatorial Theory [duplicate]

I am looking for a good book on Elementary Combinatorics (Olympiad level). For some reason I do not like Lint. I am currently reading "A Walk Through Combinatorics" by Miklos Bona and I find it really ...
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54 views

Game with matches. Very interesting mathematical problem.

Suppose you have a set of matches. You arrange them in 9 rows such that the first row has one match the second two matches the third three and so on until the ninth row which has nine matches. There ...
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76 views

A game with rice

You have $N$ rices, and K places. You can put or take a rice in place numbered $1$ at any time. You can put a rice or take a rice from a place numbered $i$ iff there is a rice at a place $i-1$. For ...
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68 views

Number of paths that lie under the diagonal

Consider a grid in $\mathbb{N}_0^2$. We can draw a path in it by traveling from point to point via a horizontal line segment to the right or vertical line segment going up. Let $k,n \in \mathbb{N}$ ...
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60 views

Game theory question- boxes

There are two players 1 and 2, and the game begins with player 1 selecting one of the boxes marked 1 to 16. Following such a selection, the selected box, as well as all boxes in the square of which ...
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on a game playable with tokens

Here is a two-player game playable with tokens. At the beginning, all tokens form a single heap. Players must choose among all heaps one of them and cut it in two parts, so that all heaps have ...
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95 views

In the card game “Projective Set”: Compute the probability that $n$ cards contain a set

In the game of Projective Set, it turns out that any seven cards contain a projective set. For fewer than 7 cards, how can we determine the probability that one or more sets exist (in terms of the ...
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120 views

In the card came “Projective Set”, show that 7 cards do always contain a set. [duplicate]

In the game of Projective Set, it turns out that any seven cards contain a projective set. How can one prove this? And for fewer than 7 cards, how can we determine the probability that one or more ...
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153 views

Wizard against two dwarfs: guess the whole function

An evil wizard plays the following game with two dwarfs $A$ and $B$: he thinks of a function $f:\mathbb{R}\to\mathbb{R}$ (which is not required to have any regularity properties, such as ...
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76 views

Sierpinski triangle game for 3 players

The players are red, green and blue. The game is played on a n-deep Sierpinski triangle. Each player colors a (black) triangle, starting at one of the main vertices. They then take turns to color an ...
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69 views

Perudo Game - probability of succes for my call

I'm making a computer program that should play as a bot against other student's bots as ICT project at school. The game is Perudo. In this part of the program I want to know what's the probability of ...
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532 views

Big List of examples of recreational finite unbounded games

What are some examples of mathematical games that can take an unbounded amount of time (a.k.a. there are starting positions such that for any number $n$, there is a line of play taking $>n$ times) ...
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NIM with multiple winning final positions

I've been looking at a variant of NIM. You can skip this bit where I'll describe NIM as usually described: There's a starting position with some number of piles of counters and two players ...
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71 views

Pseudo-Surreal numbers are analogous to?

I've been exploring surreal numbers. Real equivalent of the surreal number {0.5|} I see that pseudo-surreal numbers seem to have an interesting branch of game theory. Still having a form of {x|y}, ...
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56 views

Conway's Game OF Life maximum periods on a set x by x game board.

I have taken interest in Conway's Game of Life and want to know if you guys can help me with a mathematical problem :) That is what this website is for right? You need to be familiar with the rules ...
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79 views

Why can a Nim sum be written as powers of 2?

I have this confusion. Why do we express a nim sum as powers of 2 and why do nim sums cancel in pairs of 2 only? For instance, let's take the nim game(6,10,15) Now clearly *6 = * $2^2$ + * $2^1$ *10 ...
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128 views

Donald Knuth's Nontransitive Bingo Cards

In Time Travel and Other Mathematical Bewilderments, Martin Gardner presents a set of four nontransitive bingo cards designed by Donald Knuth (pp. 61). The rules are that the first player to complete ...
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Has this subset-sum game been studied?

Consider the following game: two players, Yolanda (who always goes first) and Zachary, take turns selecting (not yet chosen) numbers between $1$ and $9$. The first player who can make three of their ...
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267 views

Lemke-Howson pivoting in degenerate bimatrix games

I'm working on an implementation of the Lemke-Howson algorithm for finding Mixed Nash Equilibria of bimatrix games, and I'm running into a roadblock when the algorithm is fed a degenerate game. For ...
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40 views

Strategy in a blocking game

In this blocking game for 2 is there a clear strategy for the first player, the second player or neither? You have a 5 x 5 grid of squares. The players take turns laying dominoes so as to cover a ...
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N choose K and assumptions.

I have a process by which people must compare a bunch of items against each other in pairs. For now, let's say we're comparing two at a time from a set of six items. The problem is that people end up ...
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40 views

What is the probability for …?

You start at the black field bottom left and have to move op til top right black field by only move right and up. What is the probability for moving over the white field with the cross in your way to ...
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203 views

stable marriage algorithm problem

Better of the two Suppose that in the stable marriage problem with $n$ men and $n$ women, we have found two (possibly different) stable matchings $S$ and $T$. We will show how to combine $S$ and $T$ ...
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stable marriage problem doubt

Suppose we relax the rules for the men, so that each unpaired man proposes to the next woman on his list at a time of his choice (some men might procrastinate for several days, while others might ...
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Stable Marriage Algorithm Doubt

Please give me more insight on these questions and correct them if they are wrong. (True/False) In a stable marriage algorithm execution which takes n days, there is a woman who did not receive a ...
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57 views

Does Alice has a winning strategy?

There are $n$ stones, Alice and Ben are playing a game that, each one take some stones in turn, and each turn one can only take 1, 2, 4, or 6 stones, the one take the last stone wins. If Alice is the ...
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201 views

Create the most 'stressful' tennis game ever!

Some games, such as tennis, use a complicated points system (point, game, set, match; with deuces and tie-breaks) for what would otherwise be an extremely simple and monotonous game. The main reason, ...
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Explaining a Pattern in a Matrix Generated by Minimum Excluded Number in Rows & Columns

I have been given the following math puzzle: You are given a matrix that is filled by the following rule: Every cell i,j is evaluated by taking the lowest non-negative number that is not ...
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Combinatorial approach to calculate determinant

Suppose you have set of $n*n$ matrices with entries from the set $\{1,-1\}$. Then what can be the maximum determinant which you can obtain from such type of matrices.
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Is it true that we can get zero for all $(x,y,z)\in\mathbb{N}^3$?

There are three distinct positive integers $x$, $y$, and $z$. We can choose two numbers $a,b\in\{x,y,z\}$, where $b\leq a$, then replace $b$ by $2b$ and replace $a$ by $a-b$. Is it true that there ...
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Reducing an I-optimal problem to a Pareto-optimal problem

Given a set $\textbf y\subset\mathbb R^2$, let $y = (y_1,y_2), y'=(y'_1,y'_2)\in\textbf y$ be elements of that set, let $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
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Biobjective optimisation, pareto non-domination

Ok, so, I have a function $f_I(y_1, y_2) = \max\{\alpha y_1 + (1-\alpha)y_2:\alpha\in[\alpha_{min},\alpha_{max}]\}$ that I'm trying to minimise, and I'm asked to find, amongst a set of vectors $y$, ...