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Topic: Doubling the cube

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	GammonEmpire - Doubling Cube Rules
		The doubling cube is the very heart of backgammon and makes it the exciting game that it is. Without it the game might well have died off in the 1920`s.
		At the commencement of play, the doubling cube rests to one side of the board, in the centre between the two players with a displayed value of 64 (there is no 1 on the doubling cube so 64 serves as 1 at the start of the game).
		Doubling is far more complex than this because cube ownership, psychology, gammons and backgammons (and in tournaments the match score) all play a part in doubling decisions but that is beyond the scope of this first article on the topic.
	www.gammonempire.com /doubling-cube.php (593 words)

	PlanetMath: classical problems of constructibility
		Doubling the cube: Given an arbitrary cube, can a cube with double the volume be constructed?
		The discovery that doubling the cube using only compass and straightedge is impossible is also attributed to Pierre Wantzel.
		Theorem 2 (Wantzel) Doubling the cube is impossible using only compass and straightedge.
	planetmath.org /encyclopedia/SquaringTheCircle.html (553 words)

	The Backgammon Doubling Cube History
		It is unknown if the doubling cube used in the 1600s was reintroduced 250 years later or if someone came up with the idea anew after all that time had passed.
		The doubling cube itself is a simple cube with six sides similar to the dice that sits in the middle of the board during a game.
		After a player has doubled, if the opponent accepts the double, they are given possession of the cube and have the sole right to decide if the stakes should rise again during the game.
	www.justbg.com /advanced_rules_backgammon_doubling_cube.htm (545 words)

	Doubling the cube
		The origins of the problem of doubling the cube may be somewhat obscure as we have just seen, but there is no doubt that the Greeks had known for a long time how to solve the problem of doubling the square.
		Whatever the reasoning that took Hippocrates to show that the problem of doubling the cube reduced to (ii), it is quite remarkable that all later mathematicians attacked the problem (ii) rather than the original formulation.
		Although these many different methods were invented to double the cube and remarkable mathematical discoveries were made in the attempts, the ancient Greeks were never going to find the solution that they really sought, namely one which could be made with a ruler and compass construction.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Doubling_the_cube.html (2955 words)

	Doubling Rules for Online Backgammon
		Doubling is a major part of the winning online backgammon games, and it gives players the chance to catch up on their opponent even if a large gap has gathered between them, and one player is leading.
		The backgammon doubling die is a cube with the numbers 2, 4, 8, 16, 32 and 64 on its face.
		Sometimes backgammon doubling is not recommended if a gammon or backgammon could be won because it means the other opponent can decline the double and the added points would not be won on the online backgammon match.
	www.1on1backgammon.com /doubling.html (697 words)

	Schiller Institute -Pedagogy - Gauss's Fundamental Theorem of Alegebra-2
		Like the doubling of the cube, Greek geometers could not find a means for equally trisecting an arbitrary angle, from the principle of circular action itself.
		For example, the generation of square powers, corresponds to the extension that results from a doubling of the angle of rotation around the spiral; and the generation of cubed powers corresponds to the extension that results from tripling the angle of rotation.
		For example, in Figure 6, the complex number z is "squared" when the angle of rotation is doubled from x to 2 x and the length squared from A to A2.
	www.schillerinstitute.org /educ/pedagogy/gauss_fund_part2.html (2854 words)

	Doubling the cube Summary and Analysis
		It asks whether, given a cube of a certain size, it is possible to construct a cube of double the original size,...
		Doubling the cube (or duplication of the cube) is one of the three most famous geometric problems unsolvable by compass and straightedge construction.
		It was known to the Egyptians, Greeks, and Indians.[1] To double the cube means to be given a cube of...
	www.bookrags.com /Doubling_the_cube (101 words)

	Archytas (Stanford Encyclopedia of Philosophy)
		Doubling the side of the altar produces a new altar that is not twice the volume of the original altar but eight times the volume.
		The cube with which geometry deals is not a physical cube or even a drawing of a cube but rather an intelligible cube that fits the definition of the cube but is not a sense object.
		Archytas' solution to the duplication of the cube, although it was not mechanical itself, was of enormous importance for mechanics, since the solution to the problem allows one not just to double a cube but also to construct bodies that are larger or smaller than a given body in any given ratio.
	plato.stanford.edu /entries/archytas (13197 words)

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	Play Backgammon Online! Backgammon Games, Rules, Tips, and much more.
		His opponent may decline to accept the double, in which case he forfeits the game and loses 1 unit, or accept the double, in which case the game continues with the stake at 2 units.
		In money play, if player A doubles, and player B believes that he is a favorite holding the cube, he may turn the cube an extra notch as he takes, and keep the cube on his own side.
		Cube ownership remains with B. B may then if he wishes turn the cube yet another notch, saying ``aardvark'', or ``otter'' or whatever silly animal name he prefers (the correct animal is a matter of controversy), and so forth.
	www.cybercom.net /~damish/backgammon/bg-rules.html (2529 words)

	Doubling Cube \| Backgammon Rules
		The introduction of the "doubling cube" into the game in the 1920's has significantly spiced up the game, and added one more element of skill to be mastered.
		The Doubling Cube is a cube with the numbers from 2 through 64 printed on its six sides, in the sequence of powers of two.
		The game starts with the cube turned to the figure of "64" and kept at the center, between your opponent and you..The number indicates that no doubling has taken place, and the position indicates that both players have access to the cube.
	backgammon.co.uk /the-doubling-cube.html (238 words)

	M@xFriis' Doubling Home-page
		The initial double is special in the respect that you do not give your opponent the option of doubling since he has already got it (both players have value of the cube).
		This means that the doubling decision depends on the trade-off between the MWC you give up by sequences of rolls where you wish you had not doubled and your opponent's loss in MWC due to rolling a market loosing exchange after his take.
		The trailer's doubling behavior compared to money game is a little more aggressive (the leader has no value of the cube but his doubled gammon price is only 0.43).
	xfriis.dk /maxfriis/bg/double.html (6305 words)

	BGTRAINER.com
		A beaver occurs when a player is doubled but believes he is a favourite to win and he may re-double or "beaver" immediately while retaining the doubling cube and the option to re-double again later.
		Doubling Cube - A six-sided cube or die with the numbers 2, 4, 8, 16, 32 and 64 marked upon its faces used in Backgammon for raising the stakes in a game as well as to keep track of the current wager.
		The bottom of your doubling window is the minimum game-winning chance at which you are better off doubling your opponent in, than ending the game with the cube at 1.
	www.bgtrainer.com /glossary.asp (9418 words)

	Four Problems Of Antiquity from Interactive Mathematics Miscellany and Puzzles
		Construct a cube whose volume is double that of a given one.
		Doubling the Cube, the most famous of the collection, is often referred to as the Delian problem due to a legend that the Delians had consulted Plato on the subject.
		Surprisingly, the impossibility proofs for the three cited problems: doubling the cube, trisecting an angle, and constructing a regular heptagon, all fall into the same framework.
	www.cut-the-knot.org /arithmetic/antiquity.shtml (840 words)

	Hippocrates (Site not responding. Last check: )
		In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii.
		Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double.
		This had a major influence on attempts to duplicate the cube, all efforts after this being directed towards the mean proportionals problem.
	www.stetson.edu /~efriedma/periodictable/html/Cr.html (256 words)

	Gambling on backgammon with the doubling cube
		Backgammon's doubling cube is easily recognized, as it is larger than the standard backgammon dice.
		The double cube is offered by turning it over to show its next higher value and the opposing player is invited to accept it.
		Accepting the doubling cube is an agreement to continue the game at doubled stakes, while rejecting it is equivalent to resigning the game at the current stakes.
	www.gammonish.com /gambling-on-backgammon-with-the-doubling-cube-t628.html (315 words)

	Springer Online Reference Works
		The problem of constructing a cube having twice the volume of a given cube; it is one of the classical problems of Antiquity, to find an exact construction with ruler and compass.
		Like the other famous problems of quadrature of the circle and trisection of an angle, the problem of duplication of the cube belongs to the branch of geometric constructions, and is, in algebraic formulation, treated in Galois theory.
		The problem of duplication of the cube is also known as the Delian problem or the problem of doubling the cube.
	eom.springer.de /D/d034200.htm (262 words)

	Doubling Cube Backgammon - Gammon-tips.com
		Doubling Cube is a six-sided die which is not rolled, but is marked with powers of two and used to track the stakes of the current game.
		On the PartyGammon software you'll see the doubling cube on the left part of the board - Not far from it (to the left side of your screen) is the button you may use to double the value of the game.
		If the other player accepts the double, he will take control of the doubling cube and he is the only one who can make the next double offer.
	www.gammon-tips.com /doubling-cube-backgammon.html (366 words)

	The Backgammon Rules - Basic How To Play Guidelines
		One of the more well known backgammon rules is that in the event of rolling a double which means both dice show the same number the player has the ability to move four times the number shown on the dice and any combination of this.
		Doubling is an interesting concept in backgammon and one of the more interesting backgammon rules.
		The doubling cube begins in the middle of the board and either player has the option to double first.
	www.bgprime.com /backgammon-rules.html (1169 words)

	The Doubling Cube
		Doubling Cube is mysterious to many backgammon game players.
		Being 75% to win and having access to the cube is the same as being 100% to win when you can’t use the cube.
		For that matter, being 76% or 77% or 80% with the cube is the same as being 100% to win without the cube.
	www.backgammon.org /articles/the-doubling-cube (1194 words)

	Backgammon Rules
		A doubling cube, with the numerals 2, 4, 8, 16, 32, and 64 on its faces, is used to keep track of the current stake of the game.
		A player who accepts a double becomes the owner of the cube and only he may make the next double.
		The doubling cube is turned to 2 and remains in the middle.
	www.bkgm.com /rules.html (1521 words)

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