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Nim
Nim is a mathematical combinatorial game in which two players take turns removing (or "nimming") objects from distinct heaps or piles. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap or pile. Depending on the version being played, the goal of the game is either to avoid taking the last object or to take the last object. Nim is fundamental to the Sprague–Grundy theorem, which essentially says that every impartial game is equivalent to a nim game with a single pile. History Variants of nim have been played since ancient times. The game is said to have originated in China—it closely resembles the Chinese game of (), or "picking stones"—but the origin is uncertain; the earliest European references to nim are from the beginning of the 16th century. Its current name was coined by Charles L. Bouton of Harvard University, who also developed the complete theory of the game in 1901, but the orig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nimatron
The Nimatron was an electro-mechanical machine that played Nim. It was first exhibited in April–October 1940 by the Westinghouse Electric Corporation at the 1939-1940 New York World's Fair to entertain fair-goers. Conceived of some months prior by Edward Condon and built by Gerald L. Tawney and Willard A. Derr, the device was a non-programmable digital computer composed of electro-mechanical relays which could respond to players' choices in the game in a dozen different patterns. The machine, which weighed over a metric ton, displayed four lines of seven light bulbs both in front of the player and on four sides of an overhead cube. Players alternated turns with the machine in removing one or more lights from one of the rows until the lights were all extinguished. The calculations were purposely delayed to give the illusion that the machine was considering moves, and winners received a token. The reception of the machine during the fair was positive, with around 100,000 games ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nimrod (computer)
The Nimrod, built in the United Kingdom by Ferranti for the 1951 Festival of Britain, was an early computer custom-built to play Nim, inspired by the earlier Nimatron. The twelve-by-nine-by-five-foot (3.7-by-2.7-by-1.5-meter) computer, designed by John Makepeace Bennett and built by engineer Raymond Stuart-Williams, allowed exhibition attendees to play a game of Nim against an Artificial intelligence (video games), artificial intelligence. The player pressed buttons on a raised panel corresponding with lights on the machine to select their moves, and the Nimrod moved afterward, with its calculations represented by more lights. The speed of the Nimrod's calculations could be reduced to allow the presenter to demonstrate exactly what the computer was doing, with more lights showing the state of the calculations. The Nimrod was intended to demonstrate Ferranti's computer design and programming skills rather than to entertain, though Festival attendees were more interested in playing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nimber
In mathematics, the nimbers, also called Grundy numbers (not to be confused with Grundy chromatic numbers), are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with ''nimber addition'' and ''nimber multiplication'', which are distinct from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering. The nimber addition and multiplication operations are associative and commutative. Each nimber is its own additive inverse. In particular for some pairs of ordinals, their nimber sum is smaller than either addend. The minimum excludant operation is applied to sets of nimbers. Definition As a class, nimbers are indexed by ordinal numbers, and form a sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Game
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a ''position'' evolves through alternating ''moves'', each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players. However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve. Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope. Combinatorial games include we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sprague–Grundy Theorem
In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim. The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game. The Sprague–Grundy theorem and its proof encapsulate the main results of a theory discovered independently b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impartial Game
In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first. The game is played until a terminal position is reached. A terminal position is one from which no moves are possible. Then one of the players is declared the winner and the other the loser. Furthermore, impartial games are played with perfect information and no chance moves, meaning all information about the game and operations for both players are visible to both players. Impartial games include Nim, Sprouts, Kayles, Quarto, Cram, Chomp, Subtract a square, Notakto, and poset games. Go and chess are not impartial, as each player can only place or move pieces of their own color. Games such as poker, dice or dominos are not impartial games as they rely on chance. Impartial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus Theory
{{multiple issues, {{confusing, date=March 2009 {{more citations needed, date=January 2019 In the mathematical theory of games, genus theory in impartial games is a theory by which some games played under the misère play convention can be analysed, to predict the outcome class of games. Genus theory was first published in the book ''On Numbers and Games'', and later in '' Winning Ways for your Mathematical Plays'' Volume 2. Unlike the Sprague–Grundy theory for normal play impartial games, genus theory is not a complete theory for misère play impartial games. Genus of a game The genus of a game is defined using the mex (minimum excludant) of the options of a game. g+ is the grundy value or nimber of a game under the normal play convention. g- or ''λ''0 is the outcome class of a game under the misère play convention. More specifically, to find g+, *0 is defined to have g+ = 0, and all other games have g+ equal to the mex of its options. To find g−, *0 has g− = 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Last Year At Marienbad
''Last Year at Marienbad'' (), released in the United Kingdom as ''Last Year in Marienbad'', is a 1961 French New Wave avant-garde psychological drama film directed by Alain Resnais and written by Alain Robbe-Grillet. Set in a palace in a park that has been converted into a luxury hotel, the film stars Delphine Seyrig and Giorgio Albertazzi as a woman and a man who may have met the year before and may have contemplated or begun an affair, with Sacha Pitoëff as a second man who may be the woman's husband. The characters are unnamed. Plot In an ornate baroque hotel populated by wealthy individuals and couples who socialize with one another, a man approaches a woman and claims they met the previous year at a similar resort (possibly Frederiksbad, Karlstadt, Marienbad, or Baden-Salsa) and had an affair. He asserts that she responded to his request to run away together by asking him to wait a year. The woman, however, insists she has never met the man. He attempts to remind her o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferranti
Ferranti International PLC or simply Ferranti was a UK-based electrical engineering and equipment firm that operated for over a century, from 1885 until its bankruptcy in 1993. At its peak, Ferranti was a significant player in power grid systems, defense electronics, and computing, and was once a constituent of the FTSE 100 Index. The company had an extensive presence in the defense sector, manufacturing advanced cockpit displays, radar transmitters, inertial navigation systems, and avionics for military aircraft, including the Tornado fighter jet. It was a pioneer in computer technology, launching the Ferranti Mark 1 in 1951, one of the world's first commercially available computers. Ferranti's global footprint extended beyond the UK, with factories and branch plants in Australia, Canada, Singapore, Germany, and the United States. The company had a strong presence in Edinburgh, with numerous branch-plants as well as an aviation facility. Despite its eventual collapse, some ... [...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 mathematics, 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, arithmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Games Column
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, until June 1986, Gardner wrote 9 more columns, bringing his total to 297. During this period other authors wrote most of the columns. In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called " Metamagical Themas" (an anagram of "Mathematical Games"). The table below lists Gardner's columns. Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by " over in the table with a hyperlink to the cover. Other articles by Gardner Gardner wrote 5 other articles for ''Scientific American''. His flexagon article in December 1956 was in all but name the first article in the series of ''Mathematical Games'' columns and led directly to the series which began the following month. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Sum
In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. The sum game finishes when there are no moves left in either of the two parallel games, at which point (in normal play) the last player to move wins. This operation may be extended to disjunctive sums of any number of games, again by playing the games in parallel and moving in exactly one of the games per turn. It is the fundamental operation that is used in the Sprague–Grundy theorem for impartial games and which led to the field of combinatorial game theory for partisan games. Application to common games Disjunctive sums arise in games that naturally break up into components or regions that do not interact except in that each player in turn must choose just one component to play in. Examples of such games are Go, Nim, Sprouts, Domineering, the Game of the Amazo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
