Sprague-Grundy theorem

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Topic: Sprague-Grundy theorem

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	Partisan game - Wikipedia, the free encyclopedia
		Partisan games are more difficult to analyze than impartial games, as the Sprague-Grundy theorem does not apply.
		Most games are partisan; for example in chess, only one player can move the white pieces.
		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)

	Combinatorial game theory - Wikipedia, the free encyclopedia
		Nim is an impartial game for two players, and subject to the normal play condition (a player who cannot move loses) the Sprague–Grundy theorem was proved in the 1930s.
		The theorem shows that all impartial games are equivalent to heaps in nim, thus showing that major unifications are possible in games considered at a combinatorial level (in which detailed strategies matter, not just pay-offs).
		The theory introduced in the 1960s of partizan games extended the impartial theory, by relaxing the condition that a play available to one player be available to both.
	www.wikipedia.org /wiki/Combinatorial_game_theory (1524 words)

	Sprague, Washington - Encyclopedia Glossary Meaning Explanation Sprague, Washington
		Sprague was officially incorporated on November 28, 1883.
		Sprague is a city located in Lincoln County, Washington.
		The orginal Sprague, Washington article can be editet
	www.encyclopedia-glossary.com /en/Sprague-Washington.html (436 words)

	Nim article - Nim National Institute Mental Health game misère Harvard University - What-Means.com
		Nim is now used as a simple illustration of the Sprague-Grundy theorem.
		Some people have noted that turning the word NIM results in WIN, but this is probably just a coincidence.
	www.what-means.com /encyclopedia/Nim (642 words)

	semigroup.txt
		This viewpoint contrasts with the typical one in normal play: there, the Sprague-Grundy theorem says that even when we're playing a sum of games with different rules, this is no harder than finding their respective nim heap equivalents, and imagining that we're playing Nim on each of them, instead.
		The theory is an algebraic one that is motivated by the effort to generalize the well-known normal-play Sprague Grundy theory to misere play.
		We haven't really learned anything that we didn't know already from the Sprague Grundy theory.
	www.plambeck.org /oldhtml/mathematics/games/misere/semigroup.txt (1338 words)

	Nim - Wikipedia, the free encyclopedia
		Normal play Nim (or more precisely the system of nimbers) is fundamental to the Sprague-Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games.
		The theorem follows by induction on the length of the game from these two lemmata.
		This is called normal play because most games follow this convention, even though Nim usually does not.
	www.wikipedia.org /wiki/Nim (1383 words)

	\documentstyle[12pt]{article}
		Grundy values are used to determine safe and unsafe moves that either player can make.
		Grundy values can be calculated to determine the winner of the game.
		There is also a shortcut to determining the Grundy value of a board of any size- a pattern of values repeat themselves.
	www.math.uncc.edu /~hbreiter/SVSM/stevespaper.html (1614 words)

	Will Simpson
		Another way to analyze this game is by using Sprague and Grundy’s base-two arithmetic with each position having a nimber (base two number) value.
		Theorem 2: The position with pieces on (0, 0) and (x, y) is P, iff (x, y) = (0, 1), (x, y) = (1, 0), or (x, y) = (t, t), t > 2.
		Here is an example proof of the Family Theorem using strong double induction, followed by a corollary stating that the Family Theorem creates all the P positions.
	www.stetson.edu /departments/mathcs/students/research/math/will/cars.htm (3446 words)

	SPRAGUE
		Find graves of people named SPRAGUE at Find-a-Grave.com (or add one that you know).
		Search the SPRAGUE Family Message Boards at Ancestry.com (if available).
		Search the SPRAGUE Family Resource Center at RootsWeb.com (if available).
	www.worldhistory.com /surname/US/S/SPRAGUE.htm (73 words)

	1988-89 AFLB Calendar
		The theorem of Sprague and Grundy tells us that every instance of an impartial game is equivalent to an instance of the game Nim.
		These theorems were used to design algorithms for the subset-sum problem that improved the ones using dynamic programming in case of dense enough inputs.
		Using analytic number theory, he and others proved theorems characterizing the set of subset sums as a collection of arithmetic progressions with the same difference.
	theory.stanford.edu /~aflb/1988-89.html (4847 words)

	Mathematical Games
		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.
		Now the only choice left is to move in the nim heap *m, but the result is the same.
	www.math.tamu.edu /~xsun/mathgame.htm (639 words)

	Clearing up the market cycle... best Sprague-Grundy Number
		The Sprague-Grundy function for Chomp by Xinyu Sun (in work 2002...
		Grundy's Game is a special case of a Nim Game.
		in an impartial game can be assigned a Grundy number which mak...
	ascot.pl /th/Fourier5/Sprague-Grundy-Number.htm (389 words)

	hol-biblio
		Formalizing a modal logic for ccs in the HOL theorem prover.
		Theorem proving as an industrial tool for system level design.
		A comparison of HOL and ALF formalizations of a categorical coherence theorem.
	hol.sourceforge.net /hol-biblio.html (9181 words)

	The Hot Game of Nim
		According to the Sprague-Grundy theory every position in an impartial game can be assigned a Grundy number which makes it equivalent to a Nim heap of that size.
		In 1930s, R. Sprague and P. Grundy developed a theory of impartial games in which Nim played a most important role.
		Then Grundy's number g(P) of position P is determined by the Mex rule, as the smallest nimber that does not appear in the sequence g(P
	www.cut-the-knot.com /ctk/May2001.shtml (1532 words)

	putnam.95
		The local central limit theorem implies that for large n all the relevant probabilities are asymptotic, so asymptotically the probability of one of the six points is six times as likely as the single point.
		By the local central limit theorem for two dimensionsal random walks, the probabilty of a) is asymptotic to c/n where c is some constant depending only on the (co)variance of the random walk.
		Beatty's Theorem: If alpha,beta are irrational and 1/alpha + 1/beta is equal to 1, then S(alpha) and S(beta) are disjoint, and their union is {1,2,3,...}.
	www.math.niu.edu /~rusin/papers/problems-math/putnam.95 (6035 words)

	1995s.tex
		Since $\alpha$ and $\beta$ are irrational, by the one-dimensional Weil theorem, the set of points $(\{-n/\alpha\}, \{-n/\beta\}$ is dense in the set of $(x,y)$ in the unit square such that $ax + by$ is an integer.
		A result in ergodic theory (the two-dimensional version of the Weil equidistribution theorem) states that if $1,r,s$ are linearly independent over the rationals, then the set of points $(\{nr\}, \{ns\}$ is dense (and in fact equidistributed) in the unit square.
		On the other hand, suppose that such a relation does hold.
	www.libfind.unl.edu /amc/a-activities/a7-problems/putnam/1995s.tex (1211 words)

	chompnim.tex
		Of course Chomp is an impartial game, therefore each position has its own nim value by the Sprague-Grundy theorem [Sprague 1935-1936; Grundy 1939] which states that every position in an impartial game is equivalent to a Nim heap.
		The nim values are often referred as the values of the {\em Sprague-Grundy (or Grundy) function}, and the $\mathcal{P}$-positions are the ones whose value of the Grundy function are 0.
		Unfortunately, since we are fixing the top rows of the positions to be calculated, the Maple package was unable to find the generized formula in Theorem \ref{thm:nim}, although it is easy to obtain by human eyes.
	www.math.tamu.edu /~xsun/chomp/chompnim/chompnim.tex (1051 words)

	Dots-and-Boxes
		Berlekamp first presented this Dots-and-Boxes theorem to a symposium at the University of Calgary in the late 1960s.
		An improved exposition of this theorem and some of its extensions appeared in Chapter 16 of Winning Ways.
		Players at any level consistently beat players at lower levels, and do so because they understand a theorem which less sophisticated players have not yet discovered.
	math.berkeley.edu /%7eberlek/cgt/dots.html (204 words)

	byrnes.tex
		The Ultimate-Periodicity Theorem {\it guarantees} that we are bound to succeed at the end, even though some of our initial guesses may prove to be wrong.
		A theorem of Lagrange states that any quadratic irrationality has an ultimately-periodic continued fraction, and the proof also uses recurrences and the pigeon-hole principle.
		It follows that we have the ``theorem" that the set of winning positions in $2$-rowed Chomp is $ \{ [a,1] \,\, ; \,\, a \geq 0 \}$.
	www.math.rutgers.edu /~zeilberg/mamarim/mamarimTeX/byrnes.tex (2959 words)

	winner.com.ru - Define Impartial - terms standards trial is exam quality according housing the
		In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game is equivalent to a nimber.
	define-impartial.winner.com.ru (466 words)

	dots_n_boxes
		The paper concludes with a theorem on optimal strategies for the special case in which $G$ is a grid of boxes.
		The main theorem of the paper is: If $G$ is a graph with an odd number of edges, then the first player has a winning strategy.
		The main result is a reduction of difficult mathematical problems such as Fermat's Last Theorem to the predictability problem of the final fate of an initial life pattern.
	www.math.niu.edu /~rusin/uses-math/games/other/dots_n_boxes (3996 words)

	eg3.tex
		Our approach arose from studying the Nim (ie, Sprague-Grundy) values of games, while it is quite possible that Lambek and Moser were not aware of connection with combinatorial games.
		After the end of each section and before the beginning of the next section please put the line:} \void{ C. Please label your Definitions, Lemmas, Theorems, etc. in the format "{\bf Theorem x}" where x is how you see fit.
		Therefore, if $a$ is irrational, then the sequences $\lceil(1+a)m\rceil,\ m=1,2,\ldots$ and $\lceil(1+1/a)m\rceil, \ m=1,2,\ldots$ partition $\{2,3,\ldots \}$, which is equivalent to Beatty's Theorem.
	www.emis.famaf.unc.edu.ar /journals/INTEGERS/papers/eg3/eg3.tex (1352 words)

	FLoC '02 - RTA Monday July 22nd
		In the context of the rewriting calculus, we introduce and study an exception mechanism that allows us to express in a simple way rewriting strategies and that is therefore also useful for expressing theorem proving tactics.
		The proposed exception mechanism is expressed in a confluent calculus which gives the ability to simply express the semantics of the FIRST tactical and to describe in full details the expression of conditional rewriting.
		To prove our results, we use and further refine, for the case of hyperbalanced terms, some well known results concerning paths, which allow for static analysis of many fundamental properties of β-reduction.
	floc02.diku.dk /RTA/Monday.html (967 words)

	sprague grundy theorem - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "sprague grundy theorem" is defined.
		We found one dictionary with English definitions that includes the word sprague grundy theorem:
	www.onelook.com /?w=sprague-grundy+theorem (73 words)

	Math 36 -- Winter 2005 -- Homework
		Topics: examples of combinatorial games: take-away, nim, green hackenbush; how to win at nim using n and binary addition; Sprague-Grundy* theorem
		Topics: mixed and pure strategies; solving 2 by 2 matrix games; the minimax theorem
		Topics: payoff matrices for general-sum games; goal-dependent strategies; the prisoner's dilemma and variations; communicating through repeated play; natural selection of survival strategies
	math.dartmouth.edu /~m36w05/homework.html (286 words)

	USC: Academic Bulletins
		(3) (Prereq: MATH 250 or 241) Projective geometry, theorem of Desargues, conics, transformation theory, affine geometry, Euclidean geometry, non-Euclidean geometries, and topology.
		theory of the Fourier transform on the line, bandlimited functions and the Paley-Weiner theorem, Shannon-Whittacker Sampling Theorem, Riesz systems, Mallat-Meyer multiresolution analysis in Lebesgue spaces, scaling functions, wavelet constructions, wavelet representation and unconditional bases, nonlinear approximation, Riesz's factorization lemma, and Daubechies' compactly supported wavelets.
		Inversion formulas; Polya counting; combinatorial designs; minimax theorems; probabilistic methods; Ramsey theory; other topics.
	www.sc.edu /bulletin/grad/GMath.html (5582 words)

	Teorema de Sprague-Grundy
		English version: Sprague-Grundy theorem Next: Diámetro de campo de modo Up
		Fue descubierto independientemente por R. Sprague y P. Grundy.
		Para los propósitos del teorema de Sprague-Grundy, un juego es un juego two-player de la información perfecta que satisface la condición de conclusión (todos los juegos acaban: no hay líneas infinitas el juego) y del la condición normal del juego (un jugador que no puede moverse pierde).
	www.yotor.net /wiki/es/te/Teorema%20de%20SpragueGrundy.htm (502 words)

	Abstracts for Fall 2001 Meeting of NES/MAA
		This talk should be accessible to anyone who has had a course in high school geometry and thought that regular hexagons were rather pretty.
		Some simple games which turn out to be not so simple, and which lead to a lot of mathematics and a lot of fun.
		In this talk, the speaker will share some serious and lighthearted observations about puzzle and game design, and strategies that a game development company needs to consider to maximize the success of a “math” type game or puzzle.
	www.southernct.edu /organizations/nesmaa/Fall2001meetingabstracts.html (778 words)

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