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Bimagic cube

Trimagic square

Bimagic square

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	Magic cube - Wikipedia, the free encyclopedia
		In this case, the cube is classed as a simple magic cube.
		If the sums of numbers on a magic cube's broken space diagonals also equal the cube's magic number, the cube is called a pandiagonal cube.
		As in the case of magic squares, a bimagic cube has the additional property of remaining a magic cube when all of the entries are squared, a trimagic cube remains a magic cube under both the operations of squaring the entries and of cubing the entries.
	en.wikipedia.org /wiki/Magic_cube (311 words)

	Multimagic Cubes
		The sum is 130 in the 16 columns, 16 rows, 16 pillars, and in the 4 main diagonals of the cube.
		Because this cube (in degree 1) contains all magic squares possible, but they are simple magic, the cube falls into a gap in John Hendricks simple, concise, and universal set of classifications for magic hypercubes.
		A trimagic cube with all its trimagic diagonals (and of course always trimagic rows, columns, pillars and triagonals) have been constructed.
	members.shaw.ca /hdhcubes/cube_multimagic.htm (2270 words)

	cube
		A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex.
		The cube is a special kind of square prism and of triangular trapezohedron, and is dual to the octahedron.
		A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes.
	www.fact-library.com /cube.html (331 words)

	net127: Perfect Magic Cube of Order 5 Discovered
		These cubes were central symmetrical, meaning that all 13 lines of three numbers including the central number satisfied the identity x + y + 63 = 189, as well as a number of other partial magic characteristics.
		For example, a magic cube is called bimagic if it and the cube obtained by squaring each of its entries are both magic (although in this case, the squares cube is naturally no longer normal).
		Similarly, the smallest known trimagic cube is of order 64, and the smallest known perfect trimagic cube is of order 256 (Boyer, Heinz).
	net127.com /archives/000971.html (525 words)

	Magic Cubes - The Road to Perfect
		This cube is analogous to a pandiagonal magic square, but instead of being able to move a row or column to the opposite side, you may move a plane.
		(This is a diagonal cube by new classification).
		In this cube, the two main diagonals of each orthogonal (planar) array sum correctly to S. these arrays are thus simple magic squares because rows and columns are already magic as per basic requirements of a magic cube.
	members.shaw.ca /hdhcubes/cube_perfect-2.htm (5442 words)

	Self-similar Magic Cubes
		Thus the complementary cube is symmetrical to the original cube.
		This cube is symmetric around a point in the center of the cube.
		This cube is symmetrical around an axis line parallel to four of the cube faces.
	members.shaw.ca /hdhcubes/cube_selfsim.htm (772 words)

	Magic square - Wikipedia, the free encyclopedia
		If not only the main diagonals but also the broken diagonals sum to the magic constant, the result is a panmagic square.
		If raising each number to certain powers yields another magic square, the result is a bimagic, a trimagic, or, in general, a multimagic square.
		Going up in dimension results in magic cubes, magic tesseracts and other magic hypercubes.
	en.wikipedia.org /wiki/Magic_square (1488 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		sent us the first order 12 trimagic square ever found, from this square with symmetric diagonal permutation 12!!/2 = 23040 other trimagic squares can be obtained.
		The one with the smallest permutation along the main diagonal I named the "Least Diagonal Representative", any derived trimagic square will have a higher permutation of the same numbers along the main diagonal.
		The excell trimagic order 12 spreadsheet gives access to all related trimagic squares, just mention the given permutation will be enough to alert attention to a particular square within these 23040 squares
	home.wanadoo.nl /aaledewinkel/Encyclopedia/DataBase/TrimagicSquaresOrder12.html (477 words)

	MULTIMAGIE.COM - Multimagic cubes
		Since it is not necessary in the definition of a standard magic cube, the 96 diagonals of the various squares making up the bimagic cube are not bimagic.
		A trimagic cube of order 256 has also been created: it is "perfect", since all its diagonals are trimagic.
		The Général Cazalas, although smart enough to construct his 64th-order trimagic square, later failed in his attempt to construct a bimagic cube.
	cboyer.club.fr /multimagie/English/Cube.htm (843 words)

	Trimagic Square (Site not responding. Last check: 2007-10-21)
		Magic Square, the square is said to be a trimagic square.
		Trimagic squares of order 32, 64, 81, and 128 are known.
		Tarry gave a method for constructing a trimagic square of order 128, Cazalas a method for trimagic squares of orders 64 and 81, and R.
	www.math.sdu.edu.cn /mathency/math/t/t365.htm (86 words)

	MULTIMAGIE.COM - Unresolved multimagic problems
		The smallest possible trimagic square is of order 12.
		Construct the smallest possible magic cube of squares (the 16x16x16 bimagic cube, when its numbers are squared, is already a magic cube of squares)
		the case k=2 means that a perfect magic cube of order 5 -if it exists- has always 63 in its center cell, average value of integers from 1 to 125.
	cboyer.club.fr /multimagie/English/Problems.htm (1029 words)

	Clearing up the market cycle... best Tetramagic Cube (Site not responding. Last check: 2007-10-21)
		Tetramagic Cube -- from MathWorld Tetramagic Cube -- from MathWorld A tetramagic cube is a magic cube that remains magic when all its numbers are squared, cubed, and taken to the fourth power.
		Included is Hendricks order 25 bimagic of 2000, and the early bi amd trimagic cubes of Boyer in 2003.
		Trimagic Cube -- from MathWorld Trimagic Cube -- from MathWorld A trimagic cube is a magic cube that remains magic when all its numbers are both squared and cubed.
	ascot.pl /th/Fourier5/Tetramagic-Cube.htm (466 words)

	MULTIMAGIE.COM - The smallest possible bimagic cube
		The smallest bimagic cubes currently known are of order 16 constructed in January 2003.
		Using this first fact, Walter Trump has proved that any plane of a bimagic cube of order 6 would contain 6, 10, 26 or 30 even numbers, and has finally proved that it is impossible to complete a bimagic cube with such planes.
		But a tetramagic cube of order 8 or less cannot exist: there is no tetramagic series from order 3 to 8.
	cboyer.club.fr /multimagie/English/Smallestbicube.htm (819 words)

	Magic Squares
		A magic square is said to be trimagic when, in addition to the requirement for a bimagic square, the square formed by replacing every number by its cube is also a magic square.
		The sum of the cubes of the numbers in each row, each column and both diagonals is equal to 8 606 720 000.
		The image of this trimagic square is rather great (75 kB), so that you can decide, if you want to load this sensational square.
	www.magic-squares.de /gallery/multimagic/example31.html (156 words)

	Glossary
		Refers to a magic cube when the eight corners of all orders of sub-cubes contained within a cube, including wrap-around, sum to 8/m of S. I have adapted this term from Gakuho Abe’s [1] term ‘compact’ which he used to indicate that all 2x2 squares sum to 4/m of m.
		A diagonal magic cube is one where both main diagonals are correct in all planar arrays.
		For example: A perfect magic cube has all triagonals summing correctly and all magic squares contained in it are perfect (perfect is an old name for pandiagonal).
	www.geocities.com /~harveyh/glossary.htm (8481 words)

	Magic cube -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		In this case, the cube is classed as a (Click link for more info and facts about simple magic cube) simple magic cube.
		If the sums of numbers on a magic cube's (Click link for more info and facts about broken space diagonal) broken space diagonals also equal the cube's magic number, the cube is called a pandiagonal cube.
		In recent years, an alternate definition for the (Click link for more info and facts about perfect magic cube) perfect magic cube has gradually come into use.
	www.absoluteastronomy.com /encyclopedia/M/Ma/Magic_cube.htm (324 words)

	Bimagic cube - Wikipedia, the free encyclopedia
		In mathematics, a bimagic cube is a magic cube that also remains magic if all of the numbers it contains are squared.
		The only known example of a bimagic cube was given by John Hendricks in 2000; it has order 25 and magic constant 195325.
		This page was last modified 23:34, 12 March 2005.
	en.wikipedia.org /wiki/Bimagic_cube (71 words)

	MULTIMAGIE.COM - Multimagic series for cubes
		As we have seen for the smallest bimagic cube, it may be interesting, in order to try to construct a p-multimagic cube of order-n, to find all the p-multimagic series of order n, that is to say the series of n different integers, from 1 to n
		Here is a summary of the number of multimagic series for each order from 3 to 8, some of these lists being downloadable as Excel files, from 44Kb to 540Kb each.
		The bimagic and trimagic series are referenced respectively under the numbers A090653 and A092312 in the Neil Sloane's Encyclopedia of Integer Sequences, ATandT Research.
	cboyer.club.fr /multimagie/English/CubeSeries.htm (158 words)

	Multimagic Squares
		In addition, a trimagic square has the additional property that if each number in the square is multiplied by itself twice (cubed, or raised to the third power) the square is still magic.
		I strongly assume that there are trimagic squares of order 12 and I guess that at least one such square will be found in the next months.
		This trimagic square is derived from Walter's original using a spreadsheet designed by Aale de Winkel.
	www.geocities.com /~harveyh/multimagic.htm (762 words)

	AOL Search results for "Trimagic Cube"
		A trimagic cube is a magic cube that remains magic when all its numbers are...
		But both my 16th-order bimagic and the Hendrick's 25th-order bimagic cubes are non-symmetrical...
		In mathematics, a trimagic cube is a magic cube that also remains magic if all of the numbers it contains are squared or...
	aolsearch.aol.com /aol/search?query=Trimagic+Cube (279 words)

	puzzel25-2 magic squares
		The magic square of order 3 shown above is not panmagic, but the magic square of order 4 can be panmagic because the numbers in each of the four rows, four columns, and eight diagonals all could add up to an equal total.
		A magic square is called bimagic or doubly magic if it remains a magic square when each element is replaced by its square; it is called trimagic or triply magic if it remains a magic square when each element is replaced by its square and its cube.
		Here you see the grid for a normal magic square of the order of five.
	www.creativepuzzels.nl /spel/speel1/puzzel25-2.htm (455 words)

	Magic Encyclopedia index DataBase magic square, magic cube, magic tesseract, magic hypercube,
		Magic Encyclopedia index DataBase magic square, magic cube, magic tesseract, magic hypercube,
		Listing of order 2 and 3 square and cube doubling data
		Full listing of all 11 order 4 projection cubes
	home.wanadoo.nl /aaledewinkel/Encyclopedia/DataBase/DataBase.html (235 words)

	Cubo de Trimagic (Site not responding. Last check: 2007-10-21)
		En matemáticas, un cubo trimagic es un cubo mágico que también sigue siendo mágico si todos los números que contiene se ajustan o se cubican.
		No se sabe ningunos cubos trimagic hasta ahora.
		English version: Trimagic cube Next: Cubo de la magia de Semiperfect Up
	www.yotor.net /wiki/es/cu/Cubo%20de%20Trimagic.htm (64 words)

	Trimagic cube -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Trimagic cube -- Facts, Info, and Encyclopedia article
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a trimagic cube is a (Click link for more info and facts about magic cube) magic cube that also remains magic if all of the numbers it contains are squared or cubed.
		(Click link for more info and facts about Bimagic cube) Bimagic cube
	www.absoluteastronomy.com /encyclopedia/t/tr/trimagic_cube.htm (170 words)

	The Zen of Magic Squares, Circles, and Stars:
		Yet who would have guessed that in the twenty-first century, mathematicians would be studying magic squares so immense and in so many dimensions that the objects defy ordinary human contemplation and visualization?
		Readers are treated to a colorful history of magic squares and similar structures, their construction, and classification along with a remarkable variety of newly discovered objects ranging from ornate inlaid magic cubes to hypercubes.
		The new multimagic record is a pentamagic square (5-multimagic) built in 2001 by André Viricel and Christian Boyer.
	sprott.physics.wisc.edu /pickover/zenad.html (1146 words)

	magic cube
		It can be shown that if a magic cube consists of the numbers 1, 2,..., n³, then it has magic number (Sloane's A027441)
		Article: Magic cube conjures virtual reality kid's tales \| New...
		Magic Encyclopedia magic square, magic cube, magic tesseract, magic...
	www.fact-library.com /magic_cube.html (147 words)

	Trimagic square - Definition up Erdmond.Com (Site not responding. Last check: 2007-10-21)
		is a magic_square that also remains magic if all of the numbers it contains are squared or cubed.
		Trimagic_squares of orders 12, 32, 64, 81 and 128 have been discovered so far; the only known trimagic_square of order 12, given below, was found in June_2002 by German mathematician Walter_Trump.
		A copy of the license is included in the section entitled
	www.erdmond.com /Trimagic_square.html (91 words)

	Magic Squares
		Some of his large variety of inlaid magic squares, cubes, and hypercubes.
		This magic square was designed by Harry A. Sayles and published in the Monist before 1916.
		The next cyclic number (in base 10) that is capable of forming a magic square in this fashion, is n/383.
	www.geocities.com /~harveyh/magicsquare.htm (899 words)

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