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Topic: Multimagic square

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	Magic Square Encyclopedia Article @ Occasioned.org (Site not responding. Last check: 2007-10-13)
		In recreational mathematics, a magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.
		The earliest magic square of order four was found in an inscription in Khajuraho, India and in the Encyclopedia of the Brethren of Purity dating from the eleventh or twelfth century; it is also a panmagic square where, in addition to the rows, columns and main diagonals, the broken diagonals also have the same sum.
		In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with their diametrically opposite numbers.
	www.occasioned.org /encyclopedia/Magic_square (2463 words)

	Informat.io on Magic Square
		The earliest magic square of order four was found in an inscription in Khajuraho, India, dating from the eleventh or twelfth century; it is also a panmagic square where, in addition to the rows, columns and main diagonals, the broken diagonals also have the same sum.
		In the magic square of order eight, the same is done; the 16 central squares and 4 squares at each corner are retained in their places and the rest are switched.
		Squares that differ trivially by a rotation or a reflection are considered equivalent and are counted just as one (see Frénicle standard form).
	www.informat.io /?title=magic-square (1923 words)

	Science > Math > Recreations > Magic Square
		Inside mathematics, a magical square (幻方) of choose north is anorth arrangement of n² amounts within the square, such that the north totals everthing told rows, all columns, & two diagonals total to the equivalent constant.
		A earliest magic square of sequentially quartet was uncovered withinside an inscription in Khajuraho, India, dating from the eleventh or even 12th century; these are as well a panmagic square where, in addition to a rows, columns & principal diagonal, a broken diagonals likewise keep close at hand the equivalent total.
		A magical total of a square is 33, the age of Jesus at the instance of the Passion.
	magic-square.generalanswers.org (1531 words)

	Magic Square -- from Wolfram MathWorld
		A square that fails to be magic only because one or both of the main diagonal sums do not equal the magic constant is called a semimagic square.
		Squares that are magic under multiplication instead of addition can be constructed and are known as multiplication magic squares.
		Variations on magic squares can also be constructed using letters (either in defining the square or as entries in it), such as the alphamagic square and templar magic square.
	mathworld.wolfram.com /MagicSquare.html (1394 words)

	Search ScienceWorld
		Trimagic squares are also called trebly magic squares, and are 3-multimagic squares.Trimagic squares of order 12, 32, and larger are known.
		An antimagic square is an nxn array of integers from 1 to n^2 such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers.
		The square knot, also called the reef knot, is a composite knot of six crossings consisting of a knot sum of a trefoil knot and its mirror image (Rolfsen 1976, p.
	scienceworld.wolfram.com /search/index.cgi?num=&q=square&start=10 (485 words)

	Puzzle 287. Multimagic prime squares
		A magic square is bimagic (or 2-multimagic) if it remains magic after each of its numbers have been squared.
		By extension, a square is P-multimagic if it remains magic after each of its numbers have been replaced by their k-th power (for k=1, 2,..., to P).
		A 3x3 bimagic square using distinct integers (or of course prime integers) is proved to be impossible.
	www.primepuzzles.net /puzzles/puzz_287.htm (651 words)

	Trexle - Magic Square (Site not responding. Last check: 2007-10-13)
		The Magic Square Record - Includes short description of a magic square, rules for trying to break a magic square record and all the records between 1975 and now.
		Multimagic Squares - Magic squares which remain magic after their entries are raised to various powers.
		Magic squares with odd rows and colums - Describes a simple method to create magic squares with an odd number of rows and columns.
	www.trexle.com /Directory/Top/Science/Math/Recreations/Magic_Square (381 words)

	Bimagic Square -- from Wolfram MathWorld
		If replacing each number by its square in a magic square produces another magic square, the square is said to be a bimagic square.
		Bimagic squares are also called doubly magic squares, and are 2-multimagic squares.
		The first known bimagic square, constructed by Pfeffermann (1891a; left figure), had order 8 with magic constant 260 for the base square and 11180 after squaring.
	mathworld.wolfram.com /BimagicSquare.html (335 words)

	MULTIMAGIE.COM - What's a multimagic square?
		It is a square n x n sized (or n-th order) where we have succeeded in placing all the numbers from 1 to n², such that the sums of the n rows, n columns and 2 main diagonals are equal.
		A magic square is bimagic if it remains magic after each of its numbers have been squared.
		As a result, this first bimagic square is also the first multimagic square in history.
	cboyer.club.fr /multimagie/English/Definition.htm (281 words)

	Magic_Square ONLINE SHOP - Magic_Square ONLINE SHOP
		The most basic magic square is a 3x3 matrix where sum of all the rows, the columns and the two diagonal equals a constant value.
		The Magic Square Record Includes short description of a magic square, rules for trying to break a magic square record and all the records between 1975 and now.
		Multimagic Squares Magic squares which remain magic after their entries are raised to various powers.
	www.eheee.com /dir_382727_0_0___.html (393 words)

	Ivars Peterson's MathTrek - Magic Squares of Squares
		Typically, a magic square consists of a set of distinct integers arranged in the form of a square so that the numbers in each row, column, and diagonal all add up to the same total.
		The most recent developments concern magic squares in which each of the entries is a different squared number—a magic square of squares.
		The smallest magic square of squares belonging to this family (below) is not one found by Euler himself.
	www.maa.org /mathland/mathtrek_06_27_05.html (667 words)

	Bimagic square - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-13)
		In mathematics, a bimagic square is a magic square that also remains magic if all of the numbers it contains are squared.
		The first known bimagic square has order 8 and magic constant 260; it has been conjectured by Bensen and Jacoby that no nontrivial bimagic squares of order less than 8 exist.
		Hendricks was able to show in 1998 that no bimagic square of order 3 exists, save for the trivial bimagic square containing the same number nine times.
	www.godseye.com /stat/en/b/i/m/Bimagic_square.html (225 words)

	Magic Square :: Recreations
		Constructing all magic squares of order three - Mathematical approach to the generation of order three squares by applying MacMahon's partition analysis.
		MathWorld: Panmagic Square - Demonstration of the properties of a pan-magic square.
		Multimagic Squares - Squares that remain magic after entries are raised to various powers.
	science.gourt.com /Math/Recreations/Magic-Square.html (1097 words)

	Multimagic Square -- from Wolfram MathWorld
		bimagic, a 3-multimagic square is called trimagic, a 4-multimagic square is called tetramagic, a 5-multimagic square is called pentamagic, and so on.
		The first known bimagic square had order eight and was constructed by Pfefferman (1891).
		Tetramagic and pentamagic squares were constructed by Christian Boyer and André Viricel in 2001 (Boyer 2001).
	mathworld.wolfram.com /MultimagicSquare.html (142 words)

	[No title]
		is the squares sum (when numbers are raised to the k-th power)
		The kind of permutation is encountered when one studies the p-Multimagic square.
		These kinds of permutaton play part in the multiplet stacking of panmagic squares
	home.wanadoo.nl /aaledewinkel/Encyclopedia/p/Permutations.html (615 words)

	Trimagic Square -- from Wolfram MathWorld
		If replacing each number by its square or cube in a magic square produces another magic square, the square is said to be a trimagic square.
		Trimagic squares are also called trebly magic squares, and are 3-multimagic squares.
		This square, illustrated above, is the smallest possible trimagic square, since Boyer and Trump subsequently proved that a trimagic square of order less than 12 cannot exist (Boyer).
	mathworld.wolfram.com /TrimagicSquare.html (177 words)

	Magic Squares
		Magic Squares are a form of number pattern that has been around for thousands of years.
		Note that the smaller and larger numbers are mixed throughout the square, not in the outside border as they would be with a bordered magic square.
		Of course this is not a pure magic square because a consecutive series of numbers from 1 to n is not used.
	www.geocities.com /~harveyh/magicsquare.htm (899 words)

	Multimagic Squares
		However, a bimagic square has the additional property that if each number in the square is multiplied by itself (squared, or raised to the second power) the resulting row, column, and diagonal sums are also magic.
		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.
		From any magic square, there are always a family of additional squares that may be obtained by various transformations.
	www.geocities.com /~harveyh/multimagic.htm (762 words)

	Multimagic Cubes
		The sum is 130 in the 16 columns, 16 rows, 16 pillars, and in the 4 main diagonals of the cube.
		The numbers from 1 to 24 were missing, and so the square had to be finished to be filled.
		This is suggestive of the fact that both main diagonals of all 3m planar squares sum correctly to S. This is the term we will use from now on, for this type of magic cube.
	members.shaw.ca /hdhcubes/cube_multimagic.htm (2270 words)

	MultiMagic Square by Chen qinwu (Site not responding. Last check: 2007-10-13)
		Since this square has n rows, it is sufficient to divide by n in order to know the magic sum S1 of each row, column or diagonal, so:
		For the bimagic square, the sum of the squares of the first integers from 1 to N is:
		For the tetramagic, Fermat gave as early as 1636, in a letter sent to Roberval, the solution for the sum of the integers from 1 to N to the 4th-power.
	cslab.stu.edu.cn /English/Formula.asp (505 words)

	Thot / Multimagie - For the in love ones with the figures (Site not responding. Last check: 2007-10-13)
		"a magic square is a square on side N (or order N) in which one managed to place all the numbers of 1 to N squared, so that the sums of N lines, N columns and 2 diagonals are all equal.
		A magic square is known as bimagic (or -multimagic) if there remains magic after having raised with the square each one of its numbers.
		By extension, a square is known as P-multimagique if the P squares formed by replacing each number by their power K (for k=1, 2..., to P) are all magic."
	thot.cursus.edu /en/rubrique.asp?no=18253 (249 words)

	MULTIMAGIE.COM - Multimagic series
		Here is a summary of the number of multimagic series for each order from 3 to 16, some of these lists being downloadable as Excel files, from 28Kb to 800Kb each.
		And Fredrik goes further: he is the first to have computed the huge number of bimagic series of order 12, and the number of multimagic series of order 13 (excepted bimagic series of order 13).
		The bimagic, trimagic, tetramagic and pentamagic series are referenced respectively under the numbers A052457, A052458, A090037 and A106646 in the Neil Sloane's Encyclopedia of Integer Sequences, ATandT Research.
	cboyer.club.fr /multimagie/English/Series.htm (623 words)

	[No title]
		The exact sum terms merely depicts all 1-agonals and all n-agonals to sum over.
		With help of tri-digital equations this author found 80 families of order 8 bimagic squares
		224 order 9 bimagic square families where found with help of bi-digital equations by this author
	home.wanadoo.nl /aaledewinkel/Encyclopedia/m/MultiMagic.html (363 words)

	Magische Quadrateagisch (Site not responding. Last check: 2007-10-13)
		In june 2002 Walter Trump published the smallest trimgaic square of the world, which has order 12.
		It is a selfcomplement and horizontal symmetrical square, which means that two such horizontal symmetrical numbers always add to the same number 145.
		You will find a lot of information concerning the history of construction and trimagic squares in general on the webpages of Walter Trump and the multimagic pages of Christian Boyer.
	www.magic-squares.de /gallery/multimagic/example50.html (106 words)

	Math Forum: Suzanne Alejandre: Magic Squares
		Magic squares received their name because there are so many relationships between the sums of the numbers filling the squares.
		Buried in the foundation of a house to dispel evil spirits, the Double Six Magic Square at the Shaanxi History Museum contains six numbers in length and breadth, the numbers in vertical, horizontal and diagonal lines add up to 111, respectively.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /alejandre/magic.square.html (176 words)

	magic series - Multimagic series
		In multimagic series additionally the sums of certain powers of the numbers have to equal constant values.
		The algorithm used for normal magic series does not work for multimagic series.
		This discovery and all about multimagic series is described at the famous website www.multimagie.com (external link) of Christian Boyer.
	www.trump.de /magic-squares/magic-series/multi.htm (531 words)

	Multimagic squares and other puzzles at puzzled.nl
		A magic square a day keeps the trouble away… or should I have said, lets the trouble stay?
		A preprint can be found at arXiv, but is rather more difficult and technical to read than the one to be published in the AMM.
		I'm not the only one having some trouble, read an article on it.
	www.puzzled.nl (231 words)

	Science - Math - Recreations - Magic Square (Site not responding. Last check: 2007-10-13)
		A Magic Square of order n is an arrangement of the numbers from 1 to n^2 (n-squared) in an n by n matrix.
		The sum of any row, any column, or any main diagonal must be the same.
		The same idea can be extended to other shapes such as stars, cubes, circles, and so on.
	www.inter.co.yu /kategorije/Science/Math/Recreations/Magic_Square (1135 words)

	Bimagic square - Wikipedia, the free encyclopedia
		This was shown for magic squares containing the elements 1 to n
		The same holds for all lines going through the center.
		This page was last modified 04:11, 23 August 2006.
	en.wikipedia.org /wiki/Bimagic_square (208 words)

	Motocykle >> Strona gÅÃ³wna Â» Science Â» Math Â» Recreations Â» Magic Square (Site not responding. Last check: 2007-10-13)
		Magic Squares, Magic Stars and Other Patterns - Pages on magic squares, stars (original material) and miscellaneous number patterns.
		Simple Magic Square Checker - Enter numbers and display the magic square in gif or jpg formats.
		Ulearn Today : article on Magic Squares - Introduction.
	www.kawasaki.motomaniak.info /Science/Math/Recreations/Magic_Square (1358 words)

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