81. A Very, Very Magic Square A 25x25 magic square with 25 sub-squares and other properties. Maple program and PostScript files available for download. http://www.math.lsa.umich.edu/~hderksen/magic.html

A very, very magic square The following 25x25 matrix is a magic square.It has the following properties: Then entries of the square are the numbers 1,2,...,625=25x25. All its column sums, row sums and both diagonal sums are all equal to 7825=25x(1+625)/2. If the square is divided up in 25 5x5 squares, then all those little squares are magic too: they all have row, column and diagonal sums equal 1565=5x(1+625)/2. If one squares all entries in in the square, the square remains magic: all row,column and diagonal sums are equal to 3263025. Maybe you will find out even more properties.... Here is a maple program to generate this 25x25 square. There is also a postscript file magicsqr.ps and a dvi-file magicsqr.dvi of the magic square. Back to my homepage

82. Nrich.maths.org :: Mathematics Enrichment :: Magic Squares For the simple 3x3, that is order 3 magic square, trial and improvement quickly Sequentially place the numbers 1 to n 2 of the n x n magic square in the http://nrich.maths.org/1337

Skip over navigation Search by Topic Tech Help Thesaurus ... Contact Us Be part of our research! Are you really good at maths? Are you the parent of someone who is really good at maths? Are you the teacher of someone who is really good at maths? If so we'd love to involve you in our research. Case studies Are you aged between 10 and 17? Read about Dr Yi Feng's research which involves talking to students over the next two years about their mathematical experiences in and out of school, and respond to the call for participants. AskNRICH Do you use the ASKNRICH site, or have you previously? Libby Jared's research involves looking at what goes on on the ASKNRICH site and why. To help with this research please respond to the ASKNRICH survey Read more about the research we are doing here This month: Magic Squares Article Printable page Article by Del Hawley Stage: 4 and 5 A basic introduction to magic squares can be found here Magic squares have intrigued people for thousands of years and in ancient times they were thought to be connected with the supernatural and hence, magical. Today, we might still think of them as being magical, for the sum of each row, column and diagonal is a constant, the magic constant. The squares intrigued me when I found that their construction was far from easy. For the simple 3x3, that is order 3 magic square, trial and improvement quickly does the job; but for higher than order 4 magic squares a method is necessary. The problem of construction is twofold. An algorithm which works for odd order squares will not work for even order squares without the further addition of another algorithm. At least, I know of no method which will work for both odd and even orders, other than trial and improvement computer programs. For the purposes of this article, I will be considering only magic squares that are constructed using consecutive integers from 1 to

83. Magic Square — Infoplease.com Encyclopedia magic square. magic square, a square divided into parts with letters or numbers inscribed therein that, whether combined vertically, horizontally, or diagonally, form the http://www.infoplease.com/ce6/society/A0831139.html

84. Finding Magic Squares Using CCM The Chemical Casting Model (CCM) generation method is demonstrated with a Java Applet. Source code and description of algorithm included. (English/Japanese) JRE required for Applet only. http://www.kanadas.com/ccm/magic-square/index.html

Japanese version [English version] [Temporary Mirror Page (Fast!)] [Original Page (Newer?!)] Magic Squares a randomized method Introduction Magic squares of degree N is a collection of N by N columns, which contain integers from 1 to N . The sum of N integers of all the columns, all the rows, or a diagonal must be the same. A method of finding a magic square using CCM is explained here. A method of finding a magic square using CCM The applet below searches for a magic square. If you used this applet in its initial state, you can track the process by your eye in some extent. (If this applet is too large, you can use this small applet If you change the option value, which is ``medium speed (20 rps)'' in its initial state, to ``full speed,'' the computation will be done as quick as possible. (20 rps means that the rule is applied 20 times per second (rps = reductions per second). However, the real rps is less than 20.) You can start the computation again using the ``restart'' button. You probably find a different solution each time because random numbers are used, and the computation time is also different each time. If you change the option, which is set to ``swapping rule'' initially, then you can change the production rule. The rules are explained in

85. Magic Squares ROCK! Explores the physical aspects of magic squares. A mass model demonstrates unique properties associated with the moment of inertia for the square. The topographical model looks at water retention patterns in the square. http://www.knechtmagicsquare.paulscomputing.com/

86. Magic Square Definition By Babylon's Free Dictionary Definition of Magic square square divided into smaller numbered squares so that that every row and column adds up to the same total http://dictionary.babylon.com/magic square/

87. Magic Square Java Applet A documented Java applet that generates Magic Squares using a given starting number, increment, and starting location. Row, column and diagonal sums are displayed on request. Odd orders up to 7 are accepted. JRE required http://www.open.ou.nl//eyn/applets/magicsq.htm

88. The Anti-Magic Square Project Starting from the definition of an Anti-Magic Square (AMS), the article presents the structure and construction methods of the AMS. Programs written in the c-language are available for download. http://ion.uwinnipeg.ca/~vlinek/jcormie/

The Anti-Magic Square Project This web site documents my 1999 summer research on a combinatorial design called the Anti-Magic square. Anti-Magic Squares are a variation on the heavily studied and well-understood magic square. In contrast, very little seems to be known about AMSs. These pages describe what was previously known about the structure and history of the AMS and also detail new discoverys regarding their enumeration and construction. Thanks for your patience and understanding, as this page is still under construction! What is an Anti-Magic Square? An Anti-Magic Square (AMS) is an arrangement of the numbers 1 to n in a square matrix such that the row, column, and diagonal sums form a sequence of consecutive integers. The arrangement to the left is Anti-Magic because sorting the sums (numbers in black on the border) yields the sequence: 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269 This is an example of an AMS(8), or Anti-Magic Square of order 8, which comes from Madachy's Mathematical Recreations Purpose Given this definition, this research project aims to answer some of the following questions:

89. Muse: Magic Squares, Science News For Kids: MatheMUSEments of Durer s square and an alphamagic square....... http://www.sciencenewsforkids.org/pages/puzzlezone/muse/muse1103.asp

Home About Us Sponsors Article Archive ... Weather SNK RSS Feed MatheMUSEments En Español: Arte Digital For Teachers and Parents Search MatheMUSEments Magic Squares By Ivars Peterson Muse , November/December 2003, p. 32-33. Do you have a lucky number? In ancient China, people believed that a special arrangement of nine numbers in a square was especially lucky. They engraved this pattern on stones or medallions that were worn as charms to ward off evil or bring good fortune. Here's the pattern. Can you tell what's special about it? Notice it contains all the numbers from 1 to 9. Better yet, the numbers in each row, column, and diagonal add up to 15. Arrangements of numbers that add up to the same total in every row, column, and diagonal are known as magic squares. Melancholia , an engraving by the German artist Albrecht Dürer, includes a famous magic square. The rows, columns, and main diagonals all sum to 34. The magic square is hanging on the wall to the upper right. Not only do the rows, columns, and diagonals total 34, so do the numbers in the corner squares and the numbers in the central four squares. Can you find other combinations within the square that add to 34? There are several. For example, If you divide the four-by-four square into four two-by-two squares, each of those squares will add up to 34. What's more, the numbers in the middle bottom squares read 1514, the year Dürer made the engraving.

90. Mutsumi Suzuki: Magic Stars Introduction to Magic Stars. Construction, analysis and transformations. http://mathforum.org/alejandre/magic.star/

A Math Forum Web Unit Suzanne Alejandre and Mutsumi Suzuki's Magic Stars Suzanne's Math Lessons Magic Squares Tessellation Tutorials Contents What is a magic star? Magic star sets Transforming magic stars Combining exchange rules Mutsumi Suzuki is Professor of Engineering in the Laboratory for Process Systems Engineering, Tohoku University, Sendai, Japan. His research interests include reduced gravity and chemical engineering, and process system engineering. Magic Squares are one of his hobbies, of which the magic stars outlined on these pages are an extension. Home The Math Library Quick Reference Search ... Help http://mathforum.org/ The Math Forum is a research and educational enterprise of the Goodwin College of Professional Studies Mutsumi Suzuki 15 October 1996 Web page design by Sarah Seastone

91. Making Magic Squares -- Math Fun Facts A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Ever wonder how to construct a magic square? http://www.math.hmc.edu/funfacts/ffiles/10001.4-8.shtml

hosted by the Harvey Mudd College Math Department Francis Su The Math Fun Facts App is now in the App Store Subscribe to our RSS feed or Any Easy Medium Advanced The Math Fun Facts App! List All List Recent List Popular ... Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Easy level: Making Magic Squares Figure 1 Figure 2 A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Ever wonder how to construct a magic square? A silly way to make one is to put the same number in every entry of the matrix. So, let's make the problem more interesting- let's demand that we use the consecutive numbers I will show you a method that works when N is odd. As an example, consider a 3x3 magic square, as in Figure 1. Start with the middle entry of the top row. Place a 1 there. Now we'll move consecutively through the other squares and place the numbers 2, 3, 4, etc. It's easy: after placing a number, just remember to always move: 1. diagonally up and to the right when you can

92. Magic Cubes And Tesseracts Examples, original works, algorithms, and theorems on magic cubes, tesseracts, and hypercubes. (English/Japanese) http://homepage2.nifty.com/googol/magcube/en/

93. The Magic Square E 6 Up Exceptional Lie Algebras Previous F 4 4.3 The Magic Square Around 1956, Boris Rosenfeld had the remarkable idea that just as is the isometry group of the projective http://math.ucr.edu/home/baez/octonions/node16.html

Next: E Up: Exceptional Lie Algebras Previous: F 4.3 The Magic Square Around 1956, Boris Rosenfeld [ ] had the remarkable idea that just as is the isometry group of the projective plane over the octonions, the exceptional Lie groups and are the isometry groups of projective planes over the following three algebras, respectively: the bioctonions the quateroctonions the octooctonions There is definitely something right about this idea, because one would expect these projective planes to have dimensions 32, 64, and 128, and there indeed do exist compact Riemannian manifolds with these dimensions having and as their isometry groups. The problem is that the bioctonions, quateroctonions and and octooctonions are not division algebras, so it is a nontrivial matter to define projective planes over them! The situation is not so bad for the bioctonions: is a simple Jordan algebra, though not a formally real one, and one can use this to define in a manner modeled after one of the constructions of . Rosenfeld claimed that a similar construction worked for the quateroctonions and octooctonions, but this appears to be false. Among other problems, and do not become Jordan algebras under the product . Scattered throughout the literature [ ] one can find frustrated comments about the lack of a really nice construction of and . One problem is that these spaces do not satisfy the usual axioms for a projective plane. Tits addressed this problem in his theory of `buildings', which allows one to construct a geometry having any desired algebraic group as symmetries [

94. The Math Behind The Siamese Method Of Generating Magic Squares The Siamese method generates odd squares of any order above one. Explains the mathematics behind the scenes. http://www.xs4all.nl/~thospel/siamese.html

The math behind the Siamese method of generating magic squares What is a magic square A magic square consists of the distinct positive integers 1, 2, ... n , such that the sum of the n numbers in any horizontal, vertical, or main diagonal line is always the same magic constant. If the rows and columns sum to the magic constant (so ignoring the main diagonals) it is called a semimagic square. Several ways exist to automatically generate magic squares, among which the most famous one is the Siamese method to generate odd-sized magic squares. It is also called the de la Loubere's method. The method is purported to have been first reported in the West when de la Loubere returned to France after serving as ambassador to Siam. Example of the classic constructions This example is taken from the excellent description on Math Forum . I reproduce it almost literally here in case that link ever goes away. Start with an empty n x n square, where n is odd . We'll begin with n = 5 and denote its blank entries with dots.

95. Magic Square Arrange the numbers 1 through 9 in a 3 by 3 array a Magic Square such that the sum of any row, column, or the two diagonals is the same. Is your solution unique? http://jwilson.coe.uga.edu/emt725/BotCan/Magic.html

PROBLEM: Magic Square Arrange the numbers through in a 3 by 3 array a Magic Square such that the sum of any row, column, or the two diagonals is the same. Is your solution unique? That is, aside from rotation of the square, is there only one way to enter the digits? Find other 3 by 3 magic squares using distinct entries other than 1 through 9. Is it possible to complete a 3 by 3 magic square where the middle square has 21 entered in it? (Each of the other 8 squares would have a unique entry other than 21.) Can a 4 by 4 magic square be completed with the numbers 1 through 16 for entries? Find a 3 X 3 magic square where the operation is multiplication rather than addition and the entries are 9 different numbers. Return to the EMAT 6600 Page

96. Magic Cubes The site presents a program for finding all possible order 3 cubes, executable and source included. http://www.delphiforfun.org/Programs/Source_Listings/MagicCubeSource.html

97. Magic Square - Definition Of Magic Square By The Free Online Dictionary, Thesaur The kids not just quickly explain the meaning after the song but also present a Magic Square equation in the blink of an eye. http://www.thefreedictionary.com/magic square

98. The Zen Of Magic Squares, Circles, And Stars: Features reviews, information and index of Clifford A. Pickover s book. Publication focuses on historical and cultural attitudes towards the significance of the squares. http://sprott.physics.wisc.edu/pickover/zenad.html

The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions Clifford A. Pickover Princeton University Press, 2002 "A refreshing new look at a timeless topic, brimming over with ideas, littered with surprising twists. Anyone who loves numbers, anyone who enjoys puzzles, will find The Zen of Magic Squares, Circles, and Stars compulsive (and compulsory!) reading." Ian Stewart, University of Warwick Order from Amazon.com. "At first glance magic squares may seem frivolous (Ben Franklin's opinion, even as he spent countless hours studying them!), but I think that is wrong. The great nineteenth-century German mathematician Leopold Kronecker said 'God Himself made the whole numberseverything else is the work of men,' and Cliff Pickover's stimulating book hints strongly at the possibility that God may have done more with the integers than just create them. I don't believe in magic in the physical world, but magic squares come as close as we will probably ever see to being mathematical magic." - Paul J. Nahin, University of New Hampshire, author of Duelling Idiots and Other Probability Puzzlers

99. Grogono Magic Squares Home Page This is particularly true when the Magic Carpet approach is used to analyze or construct a magic square, e.g., to construct an order four magic square, four magic carpets would be http://www.grogono.com/magic/index.php

Home Page Make Your Own Choose by Size How Many? ... Grogono Home Size: Index Grogono Magic Squares Home Page Same Author: Acid-Base Animated Knots Stereo Art Revision: Features in Recent Revisions Introduction. A Magic square is intriguing; its complexity challenges the mind, certainly more than games like online bingo . For order 4 and above the number of different magic squares is astonishing - and the number remains large even if we limit consideration to Pan-Magic squares. This website reflects my own fascination with these large numbers and presents techniques aimed at explaining and reducing the huge numbers by showing how this abundance can be reduced to a small number of underlying patterns or Magic Carpets Make Your Own Pan-Magic Squares Do it yourself! This is now working again. Changes in the language had stopped it working for a while. Make your own magic square of any size up to 97x97. Discoveries. The development of this website was associated with several intriguing discoveries. Please look at the pages for the Order 4 Order 5 Order 6 magic squares.

100. Redirect Presents the Magic Carpet approach by Grog . Downloads available for analyzers and creators in Excel format. http://www.grogono.com/Magic/

Please note the correct URL: www.grogono.com/magic/index.php You will be automatically redirected in 10 seconds.

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