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Topic: Impartial games

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	Impartial game - Wikipedia, the free encyclopedia
		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.
		Impartial games can be analyzed using Sprague-Grundy theory.
		A game that is not impartial is called a partisan game.
	en.wikipedia.org /wiki/Impartial_game (118 words)

	Partisan game - Wikipedia, the free encyclopedia
		Most games are partisan; for example in chess, only one player can move the white pieces.
		Partisan games are more difficult to analyze than impartial games, as the Sprague-Grundy theorem does not apply.
		However, the application of combinatorial game theory to partisan games allows the significance of numbers as games to be seen, in a way that is not possible with impartial games.
	en.wikipedia.org /wiki/Partizan_game (118 words)

	Impartial game -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-06)
		Impartial games can be analyzed using (Click link for more info and facts about Sprague-Grundy theory) Sprague-Grundy theory.
		(A game for two players who move their 16 pieces according to specific rules; the object is to checkmate the opponent's king) Chess, however, is not impartial, since one player can only move white pieces, and the other only fl.
		A game that is not impartial is called a (Click link for more info and facts about partisan game) partisan game.
	www.absoluteastronomy.com /encyclopedia/i/im/impartial_game.htm (105 words)

	Combinatorial Games (I): The World of Piles of Stones (Site not responding. Last check: 2007-09-06)
		This game shows the usefulness of allowing heaps to grow in the analysis of the game provided the consequence of this growth is reversible.
		Thus, a + b is the game value of the size of a nim heap equivalent to playing separate games of nim on piles with a and b stones.
		Although the game sum idea is a more general framework than nim-addition, nim-addition gives the same answers as nim-sum, and, in many situations, for impartial games (standard play), we need only concern ourselves with whether a position is a P-position (nim value 0) or an N-position (nim value positive).
	80-www.ams.org.library.uor.edu /featurecolumn/archive/games5.html (1860 words)

	What are finite impartial games?
		Thus Hex is not an impartial game because I can't start using your color to color in hexes.
		In a sum of games, you may make a legal move in one sub-game, and the player who is able to make the last legal move anywhere wins.
		Thus, we say that the 2-by-4 game is equivalent to the 3-by-3, but not to the 3-by-4.
	www.stolaf.edu /people/molnar/games/impartial.html (539 words)

	Combinatorial Game Theory Background
		A SUM of such games is naturally defined as the game in which each player at his turn, may choose to make any of his legal moves on any single summand.
		Games on one side of the chasms are fruitful targets for paper-and-pencil mathematical analysis, based on an ever-growing base of general theorems.
		As I use the term, “combinatorial game theory” differs from artificial intelligence in the same primary respect that mathematics differs from engineering: the emphasis is on conclusions which are provably correct rather than on conclusions which are plausible or good enough to be acceptable in some imprecise sense.
	math.berkeley.edu /~berlek/cgt/cgt-info.html (2607 words)

	Partizan game (Site not responding. Last check: 2007-09-06)
		That means some play in the game is available to one of the two players and not to the other.
		Most games played with pieces are partizan: it is usual to have pieces that 'belong' to one side, so that one player only can move them.
		The extension of the theory to partizan games also allows the significance of numbers as games to be seen, in a way that is not possible with impartial games alone.
	www.serebella.com /encyclopedia/article-Partizan_game.html (310 words)

	Scoring (Site not responding. Last check: 2007-09-06)
		The game of Scoring is very simple: the board is a strip of several squares (parameter Heap size), on which there are placed several chips (whose number is controlled by the parameter # Counters).
		A game is impartial if, after a move, there is no way to tell which of the players made this move.
		In the theory of impartial games XOR is known under the alias of nim-sum.
	www.cut-the-knot.com /recurrence/Scoring.html (934 words)

	[No title]
		Impartial games Some of the games discussed have the property that the set of moves available in a position are not dependent on whose move it is, and the winner is the one who makes the last move.
		That is, that the nimber of the join of two games is just a function of the nimbers of the two parts.
		In general in a game A, it will be possible to make moves whose nimber is any nimber less than [A], but it will also in general be possible to make moves that have nimbers greater than [A].
	www.cs.cmu.edu /afs/cs/academic/class/15859-f01/www/notes/sleator-nim-notes.txt (1214 words)

	Two Player Mathematical Games - Combinatorial Games - Numericana
		The so-called Grundy's Game is a normal game [whichever player is unable to move loses] in which a legal move consists of splitting one row into two rows of unequal sizes (thus, rows of sizes 1 or 2 can't be split).
		Grundy's Game is another example of a game which, like Kayles, is easy to play once we work out a table of Grundy numbers.
		gave another description of the same game in terms of the moves of a chess queen allowed only to travel south and/or west [the heap sizes are the queen's cartesian coordinates, both of which are zero when the queen is at the chessboard's southwest corner].
	home.att.net /~numericana/answer/games.htm (4673 words)

	The Hot Game of Nim
		Although all impartial games have temperature of zero, I think the terminology applies nicely in that case also.
		The usual usual practice in impartial games is to call a hot position N-position (advantage to the next player) and a cold one P-position (advantage to the previous player).
		Northcott's game is one incarnation of Bogus Nim.
	www.maa.org /editorial/knot/May2001.html (1590 words)

	[No title] (Site not responding. Last check: 2007-09-06)
		Periodicity Results for Some Simple Octal Games Thane Plambeck Taking and breaking games are a class of impartial games played by removing beans from a pile and leaving the pile in zero or more parts.
		The games which have Conway codes consisting only of the octal digits 0...7 are known as octal games.
		It is conjectured [R. Guy] that all finitely-specified octal games have ultimately periodic Nim-value sequences, but the conjecture is not known to hold even for all octal games with 3 or fewer code digits.
	www.ics.uci.edu /~eppstein/cgt/octal.html (222 words)

	[No title]
		Game Theory allows the player or parties to have a better understanding of what the rules and boundaries of a given situation of games are and developing strategies for players.
		Logic has a big role in games because a player looks for patterns of the game as well as logic statements to predetermine a value in a situation, primarily what is true and what is false.
		Impartial is where each player has the same possible moves in any position of the game.
	www.saintjoe.edu /~karend/m441/RomanFinalPaper.doc (3411 words)

	Taking Games Seriously
		In the game {0}, the situation is reversed.
		First of all, it's clear that the game is the disjunctive sum of simpler games each consisting of a single fraction.
		When several games are played simultaneously, Left will try to decrease the value of a fraction as little as possible, Right will try to increase its value as little as possible.
	www.maa.org /editorial/knot/April2001.html (1936 words)

	The Hot Game of Nim
		Above and beyond the theory of the game, Nim serves as an attractive playground for mathematical reasoning, the best example of which can be found in the recently republished Excursions into Mathematics by A. Beck, M. Bleicher and D. Crowe.
		The usual practice in impartial games is to call a hot position N-position (advantage to the next player) and a cold one P-position (advantage to the previous player).
		Games of No Chance is a fine collection of articles presented at an MSRI workshop (1994) on Combinatorial Game Theory.
	www.cut-the-knot.org /ctk/May2001.shtml (1532 words)

	CA Secretary of State - Vote98 - Analysis of Proposition 5
		Card games (such as poker) can be played only if the card room does not have a stake in the outcome of the game.
		Class II gambling, however, specifically excludes all card games in which the operator has a stake in the amount wagered or the outcome of the game.
		In addition, the device must pay prizes solely in accordance with a "player's pool prize system"--defined to be a prize system where all wagers collected from players are eventually returned to the winners with no opportunity for the establishment to win.
	vote98.ss.ca.gov /VoterGuide/Propositions/5analysis.htm (1516 words)

	Mathematical Games
		We are concerned with Combinatorial Games (or Mathematical Games) consisting of two-player games with complete information (the players knows all the information about the game), no chance moves (no dice), a number of, usually finite many, positions, and the output is strictly win/lose or draw/draw.
		Notice although "tied" and "drawn" are often used indiscriminately, it is suggested that "tied" mean a game is ended but nobody is winning by the rules, whereas "drawn" mean a game will not end.
		Finding the winning move from any position in an impartial game is the same as finding the next position whose value of Sprague-Grundy function is 0, because whichever position the next player moves to, the first player can move to another position whose value of Grundy function is 0.
	www.math.tamu.edu /~xsun/mathgame.htm (639 words)

	[No title]
		This game is nearly a generalization of dots and boxes, except that each player removes one edge at his turn even if he succeeds in isolating a vertex.
		Chapter 1: Notion of partizan games, illustrated by means of a number of examples, especially Blue-Red Hackenbush, which plays a fundamental role in partizan games, analogous to that played by Nim in impartial games: In a blue-red string figure, Left removes any blue edge and all edges not connected to ground anymore.
		Impartial games: Green Hackenbush (either player may remove a green edge), Nim, Nimbers (the values of impartial games).
	www.math.niu.edu /~rusin/uses-math/games/other/dots_n_boxes (3996 words)

	G13GAM -- Game Theory -- octal games
		So the general rule is that any octal game has a theory which applies equally to a certain octal game which has heaps or rows one smaller, provided that (a) each constituent 1 is moved left; (b) each constituent 2 becomes a 3; and (b) each constituent 4 is moved right and becomes a 7.
		We see that 0.04, 0.0401, 0.007 and several other games also have 0.11337 as canonical form, and are therefore essentially the same game as 0.0423; the extra possibilities in 0.0423 have no effect [as you can see if you look at the available moves when 0.0423 is played with heaps or rows of various sizes].
		As we have seen, many of the most important impartial games can be described as octal games; but many others cannot, including Grundy's game [`split a heap into two unequal parts'] and Nim-variants in which matches are taken from two or more heaps under various conditions.
	www.maths.nottingham.ac.uk /personal/anw/G13GAM/octgam.html (1018 words)

	misere games
		Given an octal game code as input, his program directly computes a presentation of its misere quotient semigroup to heap size n=1, 2, 3, in turn, together with the associated pretending functions and outcome partitions at each heap size n.
		It's similar in spirit to David Wolfe's Gamesman's Toolkit, but with the usual normal-play impartial game simplification rules (in which every game ultimately reduces to a nim heap k) replaced by their misère versions (in which more general misère games* such as the "adder" a[4] = 2 + 2 = {3,2} occur).
		Such analyses have been notoriously difficult to come by over the years, and many games, such as Dawson's Chess (from the year 1935) are still unsolved.
	www.plambeck.org /oldhtml/mathematics/games/misere (1490 words)

	Games Gather Major Fun Accolades
		Also from Fundex, Chairs was found to be a deeply challenging dexterity game with a remarkably wide range of appeal.
		Chairs is as much fun as a solitaire game as it is a competitive or cooperative game.
		Playroom Entertainment’s (http://www.playrooment.com/) Bright Idea games is a collection of four, unique children’s card games demonstrating outstanding play value.
	www.prweb.com /releases/2005/2/prweb204969.htm (295 words)

	G13GAM -- Game Theory -- impartial hackenbush
		[General theory of impartial games:] The Nim value of a collection of disjoint components is the Nim sum of the separate components.
		Equally a move in G has a symmetric counter in -G and vice versa; this move and counter either delete the articulation points and all that sail on them in both components [giving a zero game] or in neither, in which case we can again appeal to induction and the colon principle.
		In the impartial version, the value of H will perforce be a nimber, so the component I might just as well be a `snake', a simple chain of n edges, n>=0, value *n, of the appropriate length.
	www.maths.nott.ac.uk /personal/anw/G13GAM/hack.html (1691 words)

	Games Fresh:Category Top/Games/Online (Site not responding. Last check: 2007-09-06)
		Sometimes figurines are not used at all, and sometimes a whiteboard, chalkboard or similar drawing surface is used in lieu of any figures or tokens.
		The related style of freeform role-playing is less physically oriented, and is often played at conventions.The term is also used as a name for a genre of video games that almost always lack the "role-playing" element of pen-and-paper games but borrow many gameplay elements from said games.
		These games are called CRPGs which stands for "computer role-playing games" or "console role-playing games" depending on whether the game is played on a personal computer or on a video game console.These computerized simulations have become increasingly prominent over the last two decades.
	www.games-fresh.net (491 words)

	[No title]
		One variation of Wythoff's Game, slightly easier to analyze, is to restrict the number of beans taken from a pile on any one turn to let's say three.
		Nim is the granddaddy of impartial games, and if you have spent too much time on-line you may have seen this as the "fruit game".
		It is usually, but not quite always possible to "solve" a finite impartial game by working backwards from the terminal position.
	www.stolaf.edu /people/molnar/ps/problems_games.html (1648 words)

	Games of No Chance - Cambridge University Press
		This book deals with combinatorial games, that is, games not involving chance or hidden information.
		Their study is at once old and young: though some games, such as chess, have been analyzed for centuries, the first full analysis of a nontrivial combinatorial game (Nim) only appeared in 1902.
		Other familiar games are analyzed in depth, and many exciting new games are introduced.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521646529 (638 words)

	Infinite Cyclic Impartial Games - Fraenkel, Rahat (ResearchIndex)
		We define the family of locally path-bounded digraphs, which is a class of infinite digraphs, and show that on this class it is relatively easy to compute an optimal strategy (winning or nonlosing); and realize a win, when possible, in a finite number of moves.
		Introduction We are concerned with combinatorial games, which, for our purposes here, comprise 2-player games...
		3 Partizan octal games: partizan subtraction games (context) - Fraenkel, Kotzig - 1987
	labseer.ist.psu.edu /3867.html (373 words)

	Combinatorial Game Theory
		An important distinction between this subject and classical game theory (a branch of economics) is that game players are assumed to move in sequence rather than simultanously, so there is no point in randomization or other information-hiding strategies.
		Joust, a game in which two knights use up the squares of a chessboard until one is stuck.
		In this game, a position is represented by a number, and one moves by either adding or subtracting the largest prime number less than or equal to the position.
	www.ics.uci.edu /~eppstein/cgt (1499 words)

	Amazon.com: Books: Games of No Chance (Mathematical Sciences Research Institute Publications) (Site not responding. Last check: 2007-09-06)
		The Angel and the Devil play their game on an infinite chessboard, with one square for each ordered pair of integers (x, y).
		endgame algorithm, loopy games, hyperactive kos, nim values, sowing games, annihilation games, interaction between tokens, loopfree games, tepid regions, octal games, impartial games, combinatorial game theory, chilled value, finite acyclic digraph, maximal intersecting families, interposed piece, partizan games, polynomial strategy, win classification, combinatorial games, endgame databases, normal play convention, piece endgames, nim position, max player
		I think they are very interesting and puzzling, on the border line between serious mathematics (game theory and all this stuff) and "recreational math" (like the angel problem).
	www.amazon.com /exec/obidos/tg/detail/-/0521646529?v=glance (935 words)

	Programming Games with Mathematica (Site not responding. Last check: 2007-09-06)
		Finally, there is the extensive games bibliography, compiled by Aviezri Fraenkel, which can be accessed through his home page: http://www.wisdom.weizmann.ac.il/~fraenkel/.
		Playing Games with MathematicaAn introduction to 2 person impartial games and programming them in Mathematica using the game, cram, as an illustration.
		Grundy's game Like nim, collection of heaps of counters is given; however, this time, the players take turns dividing any heap into two heaps of unequal size.
	qcpages.qc.cuny.edu /~cowen (472 words)

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