|
Pairing Strategy
In a positional game, a pairing strategy is a strategy that a player can use to guarantee victory, or at least force a draw. It is based on dividing the positions on the game-board into disjoint pairs. Whenever the opponent picks a position in a pair, the player picks the other position in the same pair. Example Consider the 5-by-5 variant of Tic-tac-toe. We can create 12 pairwise-disjoint pairs of board positions, denoted by 1,...,12 below: Note that the central element (denoted by *) does not belong to any pair; it is not needed in this strategy. Each horizontal, vertical or diagonal line contains at least one pair. Therefore the following pairing strategy can be used to force a draw: "whenever your opponent chooses an element of pair ''i'', choose the other element of pair ''i''". At the end of the game, you have an element of each winning-line. Therefore, you guarantee that the other player cannot win. Since both players can use this strategy, the game is a draw. This e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positional Game
A positional game is a kind of a combinatorial game for two players. It is described by: *Xa finite set of elements. Often ''X'' is called the ''board'' and its elements are called ''positions''. *\mathcala family of subsets of X. These subsets are usually called the ''winning-sets''. * A criterion for winning the game. During the game, players alternately claim previously-unclaimed positions, until one of the players wins. If all positions in X are taken while no player wins, the game is considered a draw. The classic example of a positional game is Tic-tac-toe. In it, X contains the 9 squares of the game-board, \mathcal contains the 8 lines that determine a victory (3 horizontal, 3 vertical and 2 diagonal), and the winning criterion is: the first player who holds an entire winning-set wins. Other examples of positional games are Hex and the Shannon switching game. For every positional game there are exactly three options: either the first player has a winning strategy, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tic-tac-toe
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) 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. Gameplay Tic-tac-toe is played on a three-by-three grid by two players, who alternately place the marks X and O in one of the nine spaces in the grid. In the following example, the first player (''X'') wins the game in seven steps: There is no universally-agreed rule as to who plays first, but in this article the convention that X plays first is used. Players soon discover that the best play from both parties leads to a draw. Hence, tic-tac-toe is often played by young children who may not have discovered the optimal strategy. Because of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maker-Breaker Game
A Maker-Breaker game is a kind of positional game. Like most positional games, it is described by its set of ''positions/points/elements'' (X) and its family of ''winning-sets'' (\mathcal- a family of subsets of X). It is played by two players, called Maker and Breaker, who alternately take previously-untaken elements. In a Maker-Breaker game, Maker wins if he manages to hold all the elements of a winning-set, while Breaker wins if he manages to prevent this, i.e. to hold at least one element in each winning-set. Draws are not possible. In each Maker-Breaker game, either Maker or Breaker has a winning strategy. The main research question about these games is which of these two options holds. Examples A classic Maker-Breaker game is Hex. There, the winning-sets are all paths from the left side of the board to the right side. Maker wins by owning a connected path; Breaker wins by owning a connected path from top to bottom, since it blocks all connected paths from left to right. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Binary Tree
In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple (''L'', ''S'', ''R''), where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary (and K-ary) trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence—a term which appears in some very old programming books, before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
