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Noughts-and-crosses
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] |
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] |
State Space Complexity
Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. Measures of game complexity State-space complexity The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game. Game tree size The game tree size is the total number of possible games that can be played: the number of leaf nodes in the game tree rooted at the game's initial position. The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a tic-tac-toe game with two X and one O on the board, this position co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three Men's Morris
Three men's morris is an abstract strategy game played on a three by three board (counting lines) that is similar to tic-tac-toe. It is also related to six men's morris and nine men's morris. A player wins by forming a mill, that is, three of their own pieces in a row. Rules 240px, A board for three men's morris. This pattern has been found carved into the roof of the temple of Kurna. Each player has three pieces. The winner is the first player to align their three pieces on a line drawn on the board. There are 3 horizontal lines, 3 vertical lines and 2 diagonal lines. The board is empty to begin the game, and players take turns placing their pieces on empty intersections. Once all pieces are placed (assuming there is no winner by then), play proceeds with each player moving one of their pieces per turn. A piece may move to any vacant point on the board, not just an adjacent one. According to ''A History of Chess'', there is an alternative version in which pieces may not mov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roman Empire
The Roman Empire ( la, Imperium Romanum ; grc-gre, Βασιλεία τῶν Ῥωμαίων, Basileía tôn Rhōmaíōn) was the post-Republican period of ancient Rome. As a polity, it included large territorial holdings around the Mediterranean Sea in Europe, North Africa, and Western Asia, and was ruled by emperors. From the accession of Caesar Augustus as the first Roman emperor to the military anarchy of the 3rd century, it was a Principate with Italia as the metropole of its provinces and the city of Rome as its sole capital. The Empire was later ruled by multiple emperors who shared control over the Western Roman Empire and the Eastern Roman Empire. The city of Rome remained the nominal capital of both parts until AD 476 when the imperial insignia were sent to Constantinople following the capture of the Western capital of Ravenna by the Germanic barbarians. The adoption of Christianity as the state church of the Roman Empire in AD 380 and the fall of the Western ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point (geometry)
In classical Euclidean geometry, a point is a primitive notion that models an exact location in space, and has no length, width, or thickness. In modern mathematics, a point refers more generally to an element of some set called a space. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy; for example, ''"there is exactly one line that passes through two different points"''. Points in Euclidean geometry Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In two-dimensional Euclidean space, a point is represented by an ordered pair (, ) of numbers, where the first number conventionally represents the horizontal and is often denoted by , and the second number conventionally represents the vertical and is often denoted by . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which points are on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane. Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine, projective, and Möbius planes), but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited to just points and lines; higher-dimensional objects (planes, solids, -spaces, conics, etc.) can be used. The study of finite structures is sometimes called finite geometry. Formal definition and terminology An incidence structure is a triple ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nd Game
A ''n''''d'' game (or ''n''''k'' game) is a generalization of the game tic-tac-toe to higher dimensions. It is a game played on a ''n''''d'' hypercube with 2 players. If one player creates a line of length ''n'' of their symbol (X or O) they win the game. However, if all ''n''''d'' spaces are filled then the game is a draw. Tic-tac-toe is the game where ''n'' equals 3 and ''d'' equals 2 (3, 2). Qubic 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 4x4x4. Players take turns pla ... is the game. The or games are trivially won by the first player as there is only one space ( and ). A game with and cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line. There are a total of winning lines in a ''n''''d'' game. See also * References Tic-tac-toe {{g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harary's Generalized Tic-tac-toe
Harary's generalized tic-tac-toe or animal tic-tac-toe is a generalization of the game tic-tac-toe, defining the game as a race to complete a particular polyomino on a square grid of varying size, rather than being limited to "in a row" constructions. It was devised by Frank Harary in March 1977, and is a broader definition than that of an m,n,k-game. Harary's generalization does not include tic-tac-toe itself, as diagonal constructions are not considered a win. Like many other two-player games, strategy stealing means that the second player can never win. All that is left to study is to determine whether the first player can win, on what board sizes he may do so, and in how many moves it will take. Results Square boards Let ''b'' be the smallest size square board on which the first player can win, and let ''m'' be the smallest number of moves in which the first player can force a win, assuming perfect play by both sides. *monomino: ''b'' = 1, ''m'' = 1 *domino: ''b'' = 2, ''m' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M,n,k-game
An ''m'',''n'',''k''-game is an abstract board game in which two players take turns in placing a stone of their color on an ''m''-by-''n'' board, the winner being the player who first gets ''k'' stones of their own color in a row, horizontally, vertically, or diagonally.J. W. H. M. Uiterwijk and H. J van der Herik, ''The advantage of the initiative'', Information Sciences 122 (1) (2000) 43-58.Jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck (2002). "Games solved: Now and in the future". Artificial Intelligence. Thus, tic-tac-toe is the 3,3,3-game and free-style gomoku is the 15,15,5-game. An ''m'',''n'',''k''-game is also called a ''k''-in-a-row game on an ''m''-by-''n'' board. The ''m'',''n'',''k''-games are mainly of mathematical interest. One seeks to find the game-theoretic value, the result of the game with perfect play. This is known as solving the game. Strategy stealing argument A standard strategy stealing argument from combinatorial game theory shows t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three Men's Morris Variant Board
3 is a number, numeral, and glyph. 3, three, or III may also refer to: * AD 3, the third year of the AD era * 3 BC, the third year before the AD era * March, the third month Books * ''Three of Them'' (Russian: ', literally, "three"), a 1901 novel by Maksim Gorky * ''Three'', a 1946 novel by William Sansom * ''Three'', a 1970 novel by Sylvia Ashton-Warner * ''Three'' (novel), a 2003 suspense novel by Ted Dekker * ''Three'' (comics), a graphic novel by Kieron Gillen. * ''3'', a 2004 novel by Julie Hilden * ''Three'', a collection of three plays by Lillian Hellman * ''Three By Flannery O'Connor'', collection Flannery O'Connor bibliography Brands * 3 (telecommunications), a global telecommunications brand ** 3Arena, indoor amphitheatre in Ireland operating with the "3" brand ** 3 Hong Kong, telecommunications company operating in Hong Kong ** Three Australia, Australian telecommunications company ** Three Ireland, Irish telecommunications company ** Three UK, British teleco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
