2
votes
2answers
35 views

Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
1
vote
1answer
52 views

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 ...
1
vote
1answer
52 views

Proving something about the Game Nim

I was reading Elementary Number Theory and Its Applications by Rosen wherein I came across the problem (located on Page 31 summarized below) Consider the Game Nim. In this game there exist a finite ...
4
votes
2answers
511 views

John Nash's Hex proof

I am reading a book on Combinatorial Game Theory that describes a proof by John Nash that Hex is a 'first player' win, but I find the proof very confusing. This proof uses a strategy-stealing ...
4
votes
0answers
97 views

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 ...
0
votes
0answers
30 views

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 ...
2
votes
1answer
53 views

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 ...
1
vote
2answers
87 views

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 ...
1
vote
0answers
37 views

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. ...
0
votes
0answers
20 views

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 ...
2
votes
0answers
284 views

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 ...
1
vote
1answer
99 views

If a game ends after finite number of moves, does it mean that at least one of the players has winning strategy?

Let us consider a game played by two players and if the game reaches some of the ending positions, one of the players is declared a winner. Let us assume that the game has to end after finitely many ...
2
votes
0answers
125 views

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, ...
1
vote
0answers
73 views

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 ...
2
votes
0answers
89 views

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 ...
0
votes
1answer
300 views

What is the probability of a $4$ appearing in the game $2048$? [closed]

I'm not sure if this is the appropriate SE, so please suggest a more appropriate website if not. I'm making a facsimile of $2048$, and I've just one question I've been unable to work out: what is the ...
4
votes
1answer
67 views

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 ...
2
votes
2answers
125 views

Winning a restricted game of Nim?

Given the following piles, find the Grundy number of the initial position and make the first move in a winning strategy given that no more than two sticks may be removed from a pile at any time. Pile ...
1
vote
1answer
59 views

Determining Grundy Numbers for an inverted takeaway game

Given the following game, I need to determine a winning strategy and find the set of positions in the kernel. I figure the best way to do so would be with Grundy numbers. Rules: The game consists ...
1
vote
1answer
57 views

Coin based subtraction game

I'm having a problem in Game Theory where I am trying to understand how a subtraction game can be interpreted by a coin based game. From my book: The problem I'm having is if I have 9 coins and the ...
4
votes
1answer
137 views

A game of Chess - Ideal Solution

I am a student of physics. I have learnt some basic group theory, and I am wondering if there is any ideal solution for a given Chess game (like solving Rubik's cube). I know the no. of permutations ...
3
votes
0answers
50 views

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 ...
2
votes
1answer
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 ...
-1
votes
1answer
114 views

A Nim game variant: Odd number [closed]

Consider the variant of Nim where the allowed moves are the removals of an odd number of stones from a heap. Who's the winner and what is the winning strategy in normal play (player unable to move ...
3
votes
2answers
132 views

Game involving tiling a 1 by n board with 1 x 2 tiles?

Consider a $1$ by $n$ tiled rectangle. You want to play a game with one opponent in which you place $1$ by $2$ "dominoes" on this rectangle. The player who places the last domino wins. Which player ...
1
vote
1answer
46 views

Prove that in an impartial Game, the P-Positions all have Sprague-Grundy Value =0

I'm looking at some work with Combinatorial Game Theory and I have currently got: (P-Position is previous player win, N-Position is next player win) Every Terminal Position is a P-Position, For ...
0
votes
1answer
85 views

Partisan/Partial Game Theory

There are enough resources available on the internet regarding "impartial" game theory. But I cannot seem to find much information regarding "partial" game theory. Can someone name some such resources ...
2
votes
2answers
331 views

What is worth of a stalk in red-blue Hackenbush??

I was studying about Red-Blue hackenbush from this link http://www.link.cs.cmu.edu/15859-s11/notes/Hackenbush.pdf http://math.ucsd.edu/~wgarner/math168a/blueredhackenbush.htm this url shows a ...
1
vote
1answer
80 views

Game of Stones - Count the ways

We are given a number of piles of stones. and we can remove two stones , where both stones come from different piles. We do this until all the piles are finished or only one pile is left as we cannot ...
1
vote
2answers
53 views

Game with two players and 120 points in total

Assume the following game: The game has two players $P_{1}$ and $P_{2}$ and 15 rounds in which they play against each other. Each round gives an amount of points equal to its number, i.e. the ...
12
votes
2answers
309 views

Determining the number of valid TicTacToe board states in terms of board dimension

I am attempting to find a closed form equation in terms of $n$, for the number of valid Tic-Tac-Toe board states (ignoring symmetry), where the board has dimension $n \times n ,\; 0 \lt n,\;n \in \Bbb ...
1
vote
3answers
502 views

Chance of Winning In Tic Tac Toe

I'm sure everyone knows how to play the game of tic-tac-toe. I have just been wondering what's the probability of winning if one player started his or her move by putting his mark in the middle?
0
votes
0answers
45 views

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
2
votes
1answer
71 views

A combinatorial game about stones

There are some piles of stones. Two players move in turn. One can remove a stone from a pile or merge two piles in a move. The player that removes the last stone wins. With the number of stones in ...
0
votes
0answers
106 views

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 ...
1
vote
0answers
56 views

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 ...
27
votes
3answers
659 views

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 ...
0
votes
0answers
82 views

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 ...
3
votes
0answers
63 views

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 ...
3
votes
2answers
112 views

probable squares in a square cake

There is a probability density function defined on the square [0,1]x[0,1]. The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area. Now Alice and Bob play a ...
4
votes
0answers
77 views

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 ...
5
votes
0answers
229 views

“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 ...
5
votes
2answers
258 views

Nim addition- binary addition without carrying

A nim addition table is essentially created by putting, in any cell, the smallest number not to the left of the cell and not above that cell in its column. However, I know for a fact that nim addition ...
3
votes
1answer
367 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 ...
0
votes
1answer
352 views

Nim Variant (Restricted removal)

Alice and Bob play the following game : There are $N$ piles of stones with $S_i$ stones in the $i$th pile. Piles are numbered from 1 to $N$. Alice and Bob play alternately, with Alice starting. In a ...
2
votes
1answer
325 views

Nim Variant (reducing by divisors)

Alice and Bob play the following game. They choose a number $N$ to play with. The rules are as follows: Alice plays first, and the two players alternate. In his/her turn, a player can subtract from ...
4
votes
3answers
143 views

Consider a card game-parity

Consider a card game where the deck consists of 63 distinct cards. The deck is created in the following manner: each card consists of some number of symbols, where no two symbols are the same. There ...
2
votes
2answers
2k views

Winning strategy for a matchstick game

There are $N$ matchsticks at the table. Two players play the game. Rules: (i) A player in his or her turn can pick $a$ or $b$ match sticks. (ii) The player who picks the last matchstick loses the ...
5
votes
1answer
187 views

Stone games again

Two players are playing a stone-picking game. There are some piles of stones. The number of stones in each pile is given. Every player takes action in turns as following rules: The one in his turn ...
3
votes
1answer
95 views

Name for a certain “product game”

Let $G,H$ be two (combinatorial impartial) games. Consider the following new game $P$: The positions are the pairs of positions of $G$ and $H$. A move in $P$ is a move in $G$, or a move in $H$, or a ...