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Octal Games
The octal games are a class of two-player games that involve removing tokens (game pieces or stones) from heaps of tokens. They have been studied in combinatorial game theory as a generalization of Nim, Kayles, and similar games. Revised and reprinted as Octal games are impartial meaning that every move available to one player is also available to the other player. They differ from each other in the numbers of tokens that may be removed in a single move, and (depending on this number) whether it is allowed to remove an entire heap, reduce the size of a heap, or split a heap into two heaps. These rule variations may be described compactly by a coding system using octal numerals. Game specification An octal game is played with tokens divided into heaps. Two players take turns moving until no moves are possible. Every move consists of selecting just one of the heaps, and either * removing all of the tokens in the heap, leaving no heap, * removing some but not all of the tokens, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Game Theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players take turns changing in defined ways or ''moves'' to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kayles
Kayles is a simple impartial game in combinatorial game theory, invented by Henry Dudeney in 1908. Given a row of imagined bowling pins, players take turns to knock out either one pin, or two adjacent pins, until all the pins are gone. Using the notation of octal games, Kayles is denoted 0.77. Rules Kayles is played with a row of tokens, which represent bowling pins. The row may be of any length. The two players alternate; each player, on his or her turn, may remove either any one pin (a ball bowled directly at that pin), or two adjacent pins (a ball bowled to strike both). Under the normal play convention, a player loses when he or she has no legal move (that is, when all the pins are gone). The game can also be played using misère rules; in this case, the player who cannot move ''wins''. History Kayles was invented by Henry Dudeney Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On Numbers And Games
''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians. Martin Gardner discussed the book at length, particularly Conway's construction of surreal numbers, in his Mathematical Games column in ''Scientific American'' in September 1976. The book is roughly divided into two sections: the first half (or ''Zeroth Part''), on numbers, the second half (or ''First Part''), on games. In the ''Zeroth Part'', Conway provides axioms for arithmetic: addition, subtraction, multiplication, division and inequality. This allows an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals. The object to which these axioms apply takes the form ... [...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 Dice (singular die or dice) are small, throwable objects with marked si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 number system, hence a true octal system might use different vocabulary. In the decimal system, each place is a power of ten. For example: : \mathbf_ = \mathbf \times 10^1 + \mathbf \times 10^0 In the octal system, each place is a power of eight. For example: : \mathbf_8 = \mathbf \times 8^2 + \mathbf \times 8^1 + \mathbf \times 8^0 By performing the calculation above in the familiar decimal system, we see why 112 in octal is equal to 64+8+2=74 in decimal. Octal numerals can be easily converted from binary representations (similar to a quaternary numeral system) by grouping consecutive binary digits into groups of three (starting from the right, for integers). For example, the binary representation for decimal 74 is 1001010. Two zeroes ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Misère Game
Misère (French for "destitution"), misere, bettel, betl, or (German for "beggar"; equivalent terms in other languages include , , ) is a bid in various card games, and the player who bids misère undertakes to win no tricks or as few as possible, usually at no trump, in the round to be played. This does not allow sufficient variety to constitute a game in its own right, but it is the basis of such trick-avoidance games as Hearts, and provides an optional contract for most games involving an auction. The term or category may also be used for some card game of its own with the same aim, like Black Peter. A misère bid usually indicates an extremely poor hand, hence the name. An open or lay down misère, or misère ouvert is a 500 bid where the player is so sure of losing every trick that they undertake to do so with their cards placed face-up on the table. Consequently, 'lay down misère' is Australian gambling slang for a predicted easy victory. In Skat, the bidding can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grundy's Game
Grundy's game is a two-player mathematical game of strategy. The starting configuration is a single heap of objects, and the two players take turn splitting a single heap into two heaps of different sizes. The game ends when only heaps of size two and smaller remain, none of which can be split unequally. The game is usually played as a '' normal play'' game, which means that the last person who can make an allowed move wins. Illustration A normal play game starting with a single heap of 8 is a win for the first player provided they start by splitting the heap into heaps of 7 and 1: player 1: 8 → 7+1 Player 2 now has three choices: splitting the 7-heap into 6 + 1, 5 + 2, or 4 + 3. In each of these cases, player 1 can ensure that on the next move he hands back to his opponent a heap of size 4 plus heaps of size 2 and smaller: player 2: 7+1 → 6+1+1 player 2: 7+1 → 5+2+1 player 2: 7+1 → 4+3+1 player 1: 6+1+1 → 4+2+1+1 player 1: 5+2+1 → 4+1+2+1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repeating Decimal
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the decimal representation of becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is , whose decimal becomes periodic at the ''second'' digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals. The infinitely repeated digit sequence is called the repetend or reptend. If the repetend is a zero, this decimal representation is called a terminating decimal rather than a repeating decimal, since the zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexapawn
Hexapawn is a deterministic two-player game invented by Martin Gardner. It is played on a rectangular board of variable size, for example on a 3×3 board or on a regular chessboard. On a board of size ''n''×''m'', each player begins with ''m'' pawns, one for each square in the row closest to them. The goal of each player is to either advance a pawn to the opposite end of the board or leave the other player with no legal moves. Hexapawn on the 3×3 board is a solved game; with perfect play, White will always lose in 3 moves (1.b2 axb2 2.cxb2 c2 3.a2 c1#). Indeed, Gardner specifically constructed it as a game with a small game tree in order to demonstrate how it could be played by a heuristic AI implemented by a mechanical computer based on Donald Michie's Matchbox Educable Noughts and Crosses Engine. A variant of this game is octopawn, which is played on a 4×4 board with 4 pawns on each side. It is a forced win for White. Rules As in chess, a pawn may be moved in two diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Rayner Dawson
Thomas Rayner Dawson (28 November 1889 – 16 December 1951) was an English chess problemist and is acknowledged as "the father of Fairy Chess". He invented many fairy pieces and new conditions. He introduced the popular fairy pieces grasshopper, nightrider, and many other fairy chess ideas. Career Dawson published his first problem, a two-mover, in 1907. His chess problem compositions include 5,320 fairies, 885 , 97 selfmates, and 138 endings. 120 of his problems have been awarded prizes and 211 honourably mentioned or otherwise commended. He cooperated in chess composition with Charles Masson Fox. Dawson was founder-editor (1922–1931) of '' The Problemist'', the journal of the British Chess Problem Society. He subsequently produced ''The Fairy Chess Review'' (1930–1951), which began as ''The Problemist Fairy Chess Supplement''. At the same time he edited the problem pages of '' The British Chess Magazine'' (1931–1951). Motivation and personality From '' The O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternary Numeral System
A quaternary numeral system is base-. It uses the digits 0, 1, 2 and 3 to represent any real number. Conversion from binary is straightforward. Four is the largest number within the subitizing range and one of two numbers that is both a square and a highly composite number (the other being 36), making quaternary a convenient choice for a base at this scale. Despite being twice as large, its radix economy is equal to that of binary. However, it fares no better in the localization of prime numbers (the smallest better base being the primorial base six, senary). Quaternary shares with all fixed- radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of these properties. Relation to other positional number systems Relation to binary and hexadecimal As wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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