# Questions tagged [combinatorial-game-theory]

Use for questions relating to Combinatorial Game Theory, terminology and concepts, particularly in regard to the mathematical properties of abstract board and card games, and game solving and solved games.

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### Is there a standard definition of "optimal losing strategy" in games with a winning strategy; Chess, Connect 4, etc?

I'm thinking about sequential, rules-based games where both players have perfect information about each other and what moves they'll be able to make (such as Chess, Connect 4, tic-tack-toe, Nim, etc.)....
291 views

### in Tic-tac-toe on an NxN board, what is the minimum goal (number in a row) that guarantees a tie?

I've been working on a Tic-Tac-Toe AI (Minimax with AB pruning). As far as I can tell, for an NxN board, player 1 can always win if the goal is to get less than N-1 in a row (for N > 4). Is there a ...
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1 vote
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### Can a Pit Card Game be Won if two players refuse to trade with another particular player?

My sister and I had an argument after I stated that I did not want to play Pit with them anymore because her husband and her son for some unknown reason refused to trade with me. Since she is always ...
5k views

### Strategy for Ultimate Tic Tac Toe

Ultimate Tic Tac Toe is played on nine tic-tac-toe boards arranged in a 3 × 3 grid. Rules: Playing on a spot inside a board, determines the next board in which the opponent must play their next move. ...
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### In two-player games with perfect information, if both players play optimally, will the game always end in a draw?

I've been running simulations of two strong agents playing Chess which always ended in a draw. Followed by this simulations, can we state that two-player perfect-information games like Go and Chess ...
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2k views

### Winning strategy for NxN Tic-Tac-Toe

In a standard 3x3 Tic-Tac-Toe board, there is a strategy to win if the opponent makes a mistake. As far as I know, the strategy goes like: Capture the centers Capture the edges Capture a square which ...
• 159
1 vote
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### How many possible starting moves are there in Abalone?

I have not found a listing or reference to the initial number of moves. I believe there are 44, but I'm not sure if this list is complete: https://github.com/void4/pyabalone/blob/master/allinitmoves....
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### Is there a formal designation for "pure placement" games?

Specifically in the context of abstract or combinatorial games. As an example, Go and Reversi are both placement games. Although mathematically distinct (Reversi requires placement in the center at ...
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### What's the least amount of pieces on an Othello board where black or white can't make a move?

Say you're following the traditional rules of Othello. What is the least number of pieces possible on the board, producing a result where one of the two players(black or white) can't make a move, ...
2k views

### Dominant strategy in chess

Is there any proof ensuring there can be no dominant strategy for chess? By dominant strategy, I mean a way of playing that always works regardless what the opponent do.
1k views

### Othello - Most number of legal moves in a given turn

Can anyone help me figure out the what the largest number of possible moves for a given turn would be in Othello/Reversi? On the first turn, there are 4 possible legal moves, the next player has 3 ...
1 vote
2k views

### What is the optimal strategy to solve Hanoi Solitaire?

I recently discovered for the first time a variation of Solitaire called Hanoi (sometimes spelled Hanoy or Hanoii). My Solitaire app says it's rated easy, but I have never solved it because I only get ...
38k views

### What's the best strategy to win a tic-tac-toe variant on a 5x5-board?

First: We assume that every player is playing at its best Rules 5x5 board Player 1 begins and puts every turn one cross “X”. Player 2 puts every turn two circles “O”. They don’t have to be next to ...
414 views

### What's the name of the theory that the player with the most options has the advantage?

A long time ago, I remember reading about a theory that sounds like this: Players A and B have access to options X and Y. Player A also has access to option Z. Player A is, at worst, equal to and, ...
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### Hex board game strategy

I was teaching myself how to play the Hex board game by reading some books. I learned how to do 2x2 and 3x3 hex games by starting at the principal diagonal. I wanted to know what the winning strategy ...
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### Is Ultimate Tic Tac Toe solved?

Ultimate Tic Tac Toe is an interesting variant of Tic Tac Toe where in each of the 9 fields, a “regular” game of Tic Tac Toe is played; see the link for details. Has this game been solved already? ...
7k views

### Maximum attainable points for a single player in a two player game of Carcassonne

Let's start this question with the following assumptions about a game of Carcassonne: It's a two player game. The game is using the current basic set of rules/tiles. edit: just to be clear, I'm using ...
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7k views

### What is the optimal strategy in Quantum Tic Tac Toe?

Quantum Tic Tac Toe is a variant of Tic-Tac-Toe with "entangled" states (there are even iPhone and Android apps). In essence, a classic naught or cross isn't placed until the 'quantum state' of the ...
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### Is Circus Maximus solvable?

Circus Maximus is a chariot racing game in which players place their 3 chariots within the starting gate, and then take turns racing around the oval course to be the first to place all 3 chariots at ...
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