1.
Star (board game)
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Star is a two-player abstract strategy board game developed by Craige Schensted. It was first published in 1983 in Games magazine and it is connection game, related to games such as Hex, Y, Havannah, and TwixT. Unlike these games, however, the result is based on a player having a final score rather than achieving a specific goal. He has since developed a more complicated version called *Star with better balance between edge and center moves, writing *Star is what those other games wanted to be. Star is played on a board of hexagonal spaces, although the board can have any size and shape, a board with unequal edges is generally used to avoid ties. Players may not place stones on the partial hexagons off the edge of the board, one player places black stones on the board, the other player places white stones. The game begins with one player placing a stone on the board, to avoid giving an advantage to the first player, a pie rule is used, allowing the second player to switch sides at that point. Players then alternate turns, placing a stone on an empty hexagon on the board, players may pass, the game is over when both players pass. At the end of the game the players count their scores, a star is a group of connected stones belonging to one player that touches at least three partial edge hexagons. The score of a star is the number of edge hexagons it touches minus two, a players score is the total of all the stars of that players color. The player with the higher score wins, for any given board, the total final score of the two players is constant

2.
*Star
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*Star is a complex abstract strategy game by Ea Ea, a designer of Y. It is a redevelopment of his earlier game Star, *Star can be played on graphs of different sizes. The three shown boards have 105,180, and 275 nodes of which 30,40, note that the edges between the five centermost nodes cross each other. Two players alternately place stones of their colour on empty nodes, the game ends when the board is filled up. Each node on the perimeter of the board counts as one peri, connected groups of one color that contain fewer than two peries are removed, with the possible peri going to the surrounding group. Each remaining group is worth the number of peries it contains minus four, the player with more points wins. Draws are decided in favour of the player owning more corners, for example, a group containing exactly two peries is worth 2−4 = −2 points. This is the same as the two peries being given to the opponent and that is, creating a group with just two peries is worthless unless it disconnects opponent groups or contains a corner. *Star is closely related to games of Hex and Y where the goal is to connect certain sides of the board to each other, on the other hand, *Star is closely related to Go in which the goal is to gather more territory than the opponent. Often survival of a group in Go is achieved by connecting it to another one, in Go, all the surrounded area is counted as territory although in practice most of the territory is gathered near the perimeter

3.
Combinatorial game theory
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Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been confined to two-player games that have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, in CGT, the moves in these and other games are represented as a game tree. CGT has a different emphasis than traditional or economic theory, which was initially developed to study games with simple combinatorial structure. Essentially, CGT has contributed new methods for analyzing game trees, for using surreal numbers. The type of games studied by CGT is also of interest in artificial intelligence, in CGT there has been less emphasis on refining practical search algorithms, but more emphasis on descriptive theoretical results. An important notion in CGT is that of the solved game, for example, tic-tac-toe is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult, for instance, in 2007 it was announced that checkers has been weakly solved—optimal play by both sides also leads to a draw—but this result was a computer-assisted proof. Other real world games are too complicated to allow complete analysis today. Applying CGT to a position attempts to determine the sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is difficult unless the game is very simple. However, a number of fall into both categories. Nim, for instance, is an instrumental in the foundation of CGT. Tic-tac-toe is still used to basic principles of game AI design to computer science students. CGT arose in relation to the theory of games, in which any play available to one player must be available to the other as well. One very important such game is nim, which can be solved completely, Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. Their results were published in their book Winning Ways for your Mathematical Plays in 1982, however, the first work published on the subject was Conways 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy, Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining

4.
Surreal number
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The surreals share many properties with the reals, including the usual arithmetic operations, as such, they form an ordered field. The surreals also contain all transfinite ordinal numbers, the arithmetic on them is given by the natural operations, research on the go endgame by John Horton Conway led to another definition and construction of the surreal numbers. Conways construction was introduced in Donald Knuths 1974 book Surreal Numbers, How Two Ex-Students Turned on to Pure Mathematics, in his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuths term, and used surreals for analyzing games in his 1976 book On Numbers and Games. In the Conway construction, the numbers are constructed in stages. Different subsets may end up defining the same number, and may define the number even if L ≠ L′. So strictly speaking, the numbers are equivalence classes of representations of form that designate the same number. In the first stage of construction, there are no previously existing numbers so the representation must use the empty set. This representation, where L and R are both empty, is called 0, subsequent stages yield forms like, =1 =2 =3 and = −1 = −2 = −3 The integers are thus contained within the surreal numbers. Similarly, representations arise like, = 1/2 = 1/4 = 3/4 so that the rationals are contained within the surreal numbers. Thus the real numbers are also embedded within the surreals, but there are also representations like = ω = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. The construction consists of three interdependent parts, the rule, the comparison rule and the equivalence rule. A form is a pair of sets of numbers, called its left set. A form with left set L and right set R is written, when L and R are given as lists of elements, the braces around them are omitted. Either or both of the left and right set of a form may be the empty set, the form with both left and right set empty is also written. The numeric forms are placed in classes, each such equivalence class is a surreal number. The elements of the left and right set of a form are drawn from the universe of the surreal numbers, equivalence Rule Two numeric forms x and y are forms of the same number if and only if both x ≤ y and y ≤ x. An ordering relationship must be antisymmetric, i. e. it must have the property that x = y only when x and y are the same object and this is not the case for surreal number forms, but is true by construction for surreal numbers

5.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers

6.
Number
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Numbers that answer the question How many. Are 0,1,2,3 and so on, when used to indicate position in a sequence they are ordinal numbers. To the Pythagoreans and Greek mathematician Euclid, the numbers were 2,3,4,5, Euclid did not consider 1 to be a number. Numbers like 3 +17 =227, expressible as fractions in which the numerator and denominator are whole numbers, are rational numbers and these make it possible to measure such quantities as two and a quarter gallons and six and a half miles. What we today would consider a proof that a number is irrational Euclid called a proof that two lengths arising in geometry have no common measure, or are incommensurable, Euclid included proofs of incommensurability of lengths arising in geometry in his Elements. In the Rhind Mathematical Papyrus, a pair of walking forward marked addition. They were the first known civilization to use negative numbers, negative numbers came into widespread use as a result of their utility in accounting. They were used by late medieval Italian bankers, by 1740 BC, the Egyptians had a symbol for zero in accounting texts. In Maya civilization zero was a numeral with a shape as a symbol. The ancient Egyptians represented all fractions in terms of sums of fractions with numerator 1, for example, 2/5 = 1/3 + 1/15. Such representations are known as Egyptian Fractions or Unit Fractions. The earliest written approximations of π are found in Egypt and Babylon, in Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 =3.1250. In Egypt, the Rhind Papyrus, dated around 1650 BC, astronomical calculations in the Shatapatha Brahmana use a fractional approximation of 339/108 ≈3.139. Other Indian sources by about 150 BC treat π as √10 ≈3.1622 The first references to the constant e were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant and it is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, the first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Eulers Mechanica. While in the subsequent years some researchers used the letter c, e was more common, the first numeral system known is Babylonian numeric system, that has a 60 base, it was introduced in 3100 B. C. and is the first Positional numeral system known

7.
Sign (mathematics)
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In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or negative is called the sign of the number, zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense, in common numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero

8.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers

9.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0

10.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing

11.
Nimber
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In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of nim heaps. They arise in a larger class of games because of the Sprague–Grundy theorem. The nimbers are the ordinal numbers endowed with a new addition and nimber multiplication. The minimum excludant operation is applied to sets of nimbers, the Sprague–Grundy theorem states that every impartial game is equivalent to a nim heap of a certain size. The value of the minimum excludant mex of a set S of ordinals is defined to be the smallest ordinal that is not an element of S, the function is useful in the definitions of nimber addition and nimber multiplication. Nimber addition can be used to calculate the size of a single nim heap equivalent to a collection of nim heaps and it is defined recursively by α ⊕ β = mex, where the mex function is defined as above. For finite ordinals, the nim-sum is easily evaluated on a computer by taking the exclusive or of the corresponding numbers. So the nim-sum is written in binary as 1001, or in decimal as 9, the only number whose XOR with α is α ⊕ β is β, and vice versa, thus α ⊕ β is excluded. Nimber multiplication is defined recursively by α β = mex, except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber additive identity is the ordinal 0, and the multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the additive inverse of the ordinal α is α itself. For all natural numbers n, the set of less than 22n form the Galois field GF of order 22n. In particular, this implies that the set of nimbers is isomorphic to the direct limit as n → ∞ of the fields GF. Just as in the case of addition, there is a means of computing the nimber product of finite ordinals. The smallest algebraically closed field of nimbers is the set of less than the ordinal ωωω. It follows that as a nimber, ωωω is transcendental over the field, the following tables exhibit addition and multiplication among the first 16 nimbers. This subset is closed under both operations, since 16 is of the form 22n

12.
Combinatorial game
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Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been confined to two-player games that have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, in CGT, the moves in these and other games are represented as a game tree. CGT has a different emphasis than traditional or economic theory, which was initially developed to study games with simple combinatorial structure. Essentially, CGT has contributed new methods for analyzing game trees, for using surreal numbers. The type of games studied by CGT is also of interest in artificial intelligence, in CGT there has been less emphasis on refining practical search algorithms, but more emphasis on descriptive theoretical results. An important notion in CGT is that of the solved game, for example, tic-tac-toe is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult, for instance, in 2007 it was announced that checkers has been weakly solved—optimal play by both sides also leads to a draw—but this result was a computer-assisted proof. Other real world games are too complicated to allow complete analysis today. Applying CGT to a position attempts to determine the sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is difficult unless the game is very simple. However, a number of fall into both categories. Nim, for instance, is an instrumental in the foundation of CGT. Tic-tac-toe is still used to basic principles of game AI design to computer science students. CGT arose in relation to the theory of games, in which any play available to one player must be available to the other as well. One very important such game is nim, which can be solved completely, Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. Their results were published in their book Winning Ways for your Mathematical Plays in 1982, however, the first work published on the subject was Conways 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy, Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining

13.
Nim
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Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, the goal of the game is to avoid being the player who must remove the last object. Variants of Nim have been played since ancient times and 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 origins of the name were never fully explained. The name is such that, when viewed from the perspective, it reads win. Nim is typically played as a game, in which the player to take the last object loses. Nim can also be played as a normal game, where the player taking the last object wins. This is called normal play because the last move is a move in most games. While all normal play impartial games can be assigned a Nim value, Only tame games can be played using the same strategy as misère nim. Nim is a case of a poset game where the poset consists of disjoint chains. At the 1940 New York Worlds Fair Westinghouse displayed a machine, the Nimatron and it was also one of the first ever electronic computerized games. Ferranti built a Nim playing computer which was displayed at the Festival of Britain in 1951. In 1952 Herbert Koppel, Eugene Grant and Howard Bailer, engineers from the W. L. Maxon Corporation, developed a machine weighing 50 pounds which played Nim against a human opponent, a Nim Playing Machine has been described made from TinkerToy. The game of Nim was the subject of Martin Gardners February 1958 Mathematical Games column in Scientific American, a version of Nim is played—and has symbolic importance—in the French New Wave film Last Year at Marienbad. The normal game is between two players and played with three heaps of any number of objects, the two players alternate taking any number of objects from any single one of the heaps. The goal is to be the last to take an object, in misère play, the goal is instead to ensure that the opponent is forced to take the last remaining object. The following example game is played between fictional players Bob and Alice who start with heaps of three, four and five objects,022 Alice takes 1 from B012 Bob takes 1 from C leaving two 1s. 011 Alice takes 1 from B001 Bob takes entire C heap, Only the last move changes between misere and normal play. * Only valid for normal play, ** Only valid for misere, for the generalisations, n and m can be any value >0, and they may be the same

14.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

15.
Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field