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Kaplansky's Game
Kaplansky's game or Kaplansky's ''n''-in-a-line is an abstract board game in which two players take turns in placing a stone of their color on an infinite lattice board, the winner being the player who first gets ''k'' stones of their own color on a line which does not have any stones of the opposite color on it. It is named after Irving Kaplansky. General results * ''k'' ≤ 3 is a first-player win. * 4 ≤ ''k'' ≤ 7 is believed to be draw, but this remains unproven. * ''k'' ≥ 8 is a draw: Every player can draw via a "pairing strategy" or other "draw strategy" of ''m'',''n'',''k''-game. See also * ''m'',''n'',''k''-game * Hex (board game) * Harary's generalized tictactoe 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" construction ... References {{Tic-Tac-Toe Abstract strategy game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Game
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games need not be conceptually intricate to involve deeper computational underpinnings. For example, even though the rules of Mancala are relatively basic, the game can be rigorously analyzed through the lens of combinatorial game theory. Mathematical games differ sharply from mathematical puzzles in that mathematical puzzles require specific mathematical expertise to complete, whereas mathematical games do not require a deep knowledge of mathematics to play. Often, the arithmetic core of mathematical games is not readily apparent to players untrained to note the statistical or mathematical aspects. Some mathematical games are of deep interest in the field of recreational mathematics. When studying a game's core mathematics, arithmetic theory i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Board Game
Board games are tabletop games that typically use . These pieces are moved or placed on a pre-marked board (playing surface) and often include elements of table, card, role-playing, and miniatures games as well. Many board games feature a competition between two or more players. To show a few examples: in checkers (British English name 'draughts'), a player wins by capturing all opposing pieces, while Eurogames often end with a calculation of final scores. '' Pandemic'' is a cooperative game where players all win or lose as a team, and peg solitaire is a puzzle for one person. There are many varieties of board games. Their representation of real-life situations can range from having no inherent theme, such as checkers, to having a specific theme and narrative, such as ''Cluedo''. Rules can range from the very simple, such as in snakes and ladders; to deeply complex, as in ''Advanced Squad Leader''. Play components now often include custom figures or shaped counters, and distin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews. http://www-history.mcs.st-andrews.ac.uk/Biographies/Kaplansky.html. Biography Kaplansky or "Kap" as his friends and colleagues called him was born in Toronto, Ontario, Canada, to Polish-Jewish immigrants; his father worked as a tailor, and his mother ran a grocery and, eventually, a chain of bakeries. He went to Harbord Collegiate Institute receiving the Prince of Wales Scholarship as a teenager. He attended the University of Toronto as an undergraduate and finished first in his class for three consecutive years. In his senior year, he competed in the first William Lowell Putnam Mathematical Competition, becoming one of the first five recipients of the Putnam Fellowship, which paid for graduate studies at Harvard University. Administe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...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] |
Hex (board Game)
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. It is traditionally played on an 11×11 rhombus board, although 13×13 and 19×19 boards are also popular. The board is composed of hexagons called ''cells'' or ''hexes''. Each player is assigned a pair of opposite sides of the board, which they must try to connect by alternately placing a stone of their color onto any empty hex. Once placed, the stones are never moved or removed. A player wins when they successfully connect their sides together through a chain of adjacent stones. Draws are impossible in Hex due to the topology of the game board. Despite the simplicity of its rules, the game has deep strategy and sharp tactics. It also has profound mathematical underpinnings. The game was first published under the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harary's Generalized Tictactoe
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] |
Abstract Strategy Games
Abstract strategy games admit a number of definitions which distinguish these from strategy games in general, mostly involving no or minimal narrative theme, outcomes determined only by player choice (with no randomness), and perfect information. For example, Go is a pure abstract strategy game since it fulfills all three criteria; chess and related games are nearly so but feature a recognizable theme of ancient warfare; and Stratego is borderline since it is deterministic, loosely based on 19th-century Napoleonic warfare, and features concealed information. Definition Combinatorial games have no randomizers such as dice, no simultaneous movement, nor hidden information. Some games that do have these elements are sometimes classified as abstract strategy games. (Games such as '' Continuo'', Octiles, '' Can't Stop'', and Sequence, could be considered abstract strategy games, despite having a luck or bluffing element.) A smaller category of abstract strategy games manages to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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