Gomoku, also called Five in a Row, is an abstract strategy board game. It is traditionally played with Go pieces on a 15×15 Go board while in the past a 19×19 board was standard. Because pieces are typically not moved or removed from the board, gomoku may also be played as a paper-and-pencil game. The game is known in several countries under different names.
Pente is an abstract strategy board game for two or more players, created in 1977 by Gary Gabrel. A member of the m,n,k game family, Pente stands out for its custodial capture mechanic, which allows players to "sandwich" pairs of stones and capture them by flanking them on either side. This changes the overall tactical assessments players face when compared to pure placement m,n,k games such as Gomoku.
Tic-tac-toe, noughts and crosses, or Xs and Os is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with X or O. The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row is the winner. It is a solved game, with a forced draw assuming best play from both players.
Hex is a two player abstract strategy board game in which players attempt to connect opposite sides of a rhombus-shaped board made of hexagonal cells. Hex was invented by mathematician and poet Piet Hein in 1942 and later rediscovered and popularized by John Nash.
Fanorona is a strategy board game for two players. The game is indigenous to Madagascar.
A solved game is a game whose outcome can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory and/or computer assistance.
Connect Four is a game in which the players choose a color and then take turns dropping colored tokens into a six-row, seven-column vertically suspended grid. The pieces fall straight down, occupying the lowest available space within the column. The objective of the game is to be the first to form a horizontal, vertical, or diagonal line of four of one's own tokens. It is therefore a type of m,n,k-game with restricted piece placement. Connect Four is a solved game. The first player can always win by playing the right moves.
Combinatorial game theory measures game complexity in several ways:
- State-space complexity,
- Game tree size,
- Decision complexity,
- Game-tree complexity,
- Computational complexity.
Renju is a professional variant of gomoku. It was named renju by Japanese journalist Ruikou Kuroiwa (黒岩涙香) on December 6, 1899, in a Japanese newspaper Yorozu chouhou (萬朝報). The name "renju" means "connected pearls" in Japanese. The game is played with black and white stones on a 15×15 gridded go board.
3D tic-tac-toe, also known by the trade name Qubic, is an abstract strategy board game, generally for two players. It is similar in concept to traditional tic-tac-toe but is played in a cubical array of cells, usually 4×4×4. Players take turns placing their markers in blank cells in the array. The first player to achieve four of their own markers in a row wins. The winning row can be horizontal, vertical, or diagonal on a single board as in regular tic-tac-toe, or vertically in a column, or a diagonal line through four boards.
In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy. The strategy-stealing argument applies to any symmetric game in which an extra move can never be a disadvantage. A key property of a strategy-stealing argument is that it proves that the first player can win the game without actually constructing such a strategy. So, although it might prove the existence of a winning strategy, the proof gives no information about what that strategy is.
Domineering is a mathematical game that can be played on any collection of squares on a sheet of graph paper. For example, it can be played on a 6×6 square, a rectangle, an entirely irregular polyomino, or a combination of any number of such components. Two players have a collection of dominoes which they place on the grid in turn, covering up squares. One player places tiles vertically, while the other places them horizontally. As in most games in combinatorial game theory, the first player who cannot move loses.
Connect6 introduced in 2003 by Professor I-Chen Wu at Department of Computer Science and Information Engineering, National Chiao Tung University in Taiwan, is a two-player strategy game similar to Gomoku.
Zillions of Games is a commercial general game playing system developed by Jeff Mallett and Mark Lefler in 1998. The game rules are specified with S-expressions, Zillions rule language. It was designed to handle mostly abstract strategy board games or puzzles. After parsing the rules of the game, the system's artificial intelligence can automatically play one or more players. It treats puzzles as solitaire games and its AI can be used to solve them.
Cram is a mathematical game played on a sheet of graph paper. It is the impartial version of Domineering and the only difference in the rules is that players may place their dominoes in either orientation, but it results in a very different game. It has been called by many names, including "plugg" by Geoffrey Mott-Smith, and "dots-and-pairs". Cram was popularized by Martin Gardner in Scientific American.
In combinatorial game theory, a two-player deterministic perfect information turn-based game is a first-player-win if with perfect play the first player to move can always force a win. Similarly, a game is second-player-win if with perfect play the second player to move can always force a win. With perfect play, if neither side can force a win, the game is a draw.
Proof-number search is a game tree search algorithm invented by Victor Allis, with applications mostly in endgame solvers, but also for sub-goals during games.
Solving chess consists of finding an optimal strategy for the game of chess; that is, one by which one of the players can always force a victory, or either can force a draw. It is also related to more generally solving chess-like games such as Capablanca chess and infinite chess. In a weaker sense, solving chess may refer to proving which one of the three possible outcomes is the result of two perfect players, without necessarily revealing the optimal strategy itself.
Gomocup is a worldwide tournament of artificial intelligences (AI) playing Gomoku and Renju. The tournament has been played since 2000 and takes place every year. As of 2016, it is the most famous and largest Gomoku AI tournament in the world, with around 40 participants from about 10 countries.
Tic-tac-toe is an instance of an m,n,k-game, where two players alternate taking turns on an m×n board until one of them gets k in a row. Harary's generalized tic-tac-toe is an even broader generalization. The game can also be generalized as a nd game. The game can be generalised even further from the above variants by playing on an arbitrary hypergraph where rows are hyperedges and cells are vertices.
