# Tagged Questions

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### 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 ...
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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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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...