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