41. Magic Squares Applet Java applet and source files.JRE to run http://mathforum.org/alejandre/java/magpuz/MagPuz.html

Magic Squares Applet Note: Mike Morton wrote this applet while participating in the Math Forum's 1996 Summer Institute. Once we cataloged this applet in our Math Tools project, its functionality in different browsers and platforms was raised in a discussion. If this version does not work with your platform and/or browser, try this version of the Magic Squares Applet revised by Pavel Safronov and Michael McKelvey. Find accompanying lessons and student activities here The Source Files.... MagPuz.java MagCan.java MagControls.java ColorsWindow.java If you have any questions/suggestions, please send them to Suzanne Alejandre

42. Maths Index Articles and images on recreational math from fractals and magic squares to mathemorchids and Galois. http://www.geocities.com/aladgyma/articles/scimaths/maths.htm

Maths Index Mathemorchids The algorithms of nature. Polygrammagick The lesser-known cousins of magic squares. Reciprocal Arrangements Magic squares from the digits of prime reciprocals. Rep It Up More magic squares from reciprocals. Dying Jungian Rimbaud vs Galois, or Poetry vs Maths Platonic Hex (NEW 22/ii/2004) Magic Platonic polyhedra. Sequence and Pearls (NEW 14/iii/2004) Patterns created by modified Fibonacci sequences. The Joy of x (NEW 21/iii/2004) Patterns created by (x + y ) mod q (may load slowly). Squareway to Heaven (NEW 14/iii/2004) Magic squares dedicated to the seven heavenly bodies of traditional astrology. 3x3 Font (NEW 18/ii/2004) A minimal font. Gamma, Gamma, Gamma, Hey! The swastika and 1/273. Koch Doodles Fun with fractal snowflakes. 222 Divine Some interesting facts associated with the second repunit. 111 Chariots The 111 instances of the word "chariots" in the Old Testament of the King James Bible. Sum Beastliness Some simple mathematics associated with 666, the Number of the Beast. Delete and Repeat till Complete Deletion algorithms and multiplicative palindromes. Unless otherwise stated, all articles are by

43. Magic Squares An introduction, with algorithms for generating odd, and doubly even (4k) squares. References. http://www.jcu.edu/math/vignettes/magicsquares.htm

Vignette 20 Magic Squares An normal magic square is an arrangement of the numbers 1, 2, 3, ... n in a square array, with the property that the sum of every row and column, as well as both diagonals, is the same number. An example of a normal magic square is You can verify that each of the three rows, the three columns, and the two diagonals add to 15. Magic Squares in History Magic squares have been studied for at least three thousand years, the earliest recorded appearance dating to about 2200 BC, in China. In the 9 th magic square with some very interesting properties is attributed to him. What is the Magic Sum? Given an normal magic square, suppose M is the number that each row, column and diagonal must add up to. Then since there are n rows the sum of all the numbers in the magic square must be . But the numbers being added are 1, 2, 3, ... n , and so 1 + 2 + 3 + ... + n . In summation notation, . Using the formula for this sum, we have , and then solving for M gives . Thus, a normal magic square must have its rows, columns and diagonals adding to

44. Index Of Prime Magic Squares Introduction. Prime magic squares, programs and examples. http://digilander.libero.it/ice00/magic/

Index of Prime Magic Squares Index Magic Square: What is a magic square? Prime Magic Square: What is a prime magic square? Prime Analysis: How is the magic constant for prime magic square and what order are available? Muncey Analysis: How was build the Muncey order 12 magic square? Manual Construction: see how I manually (with computer help) construct squares of order 15 and 17. Automatic Construction: see how my programs build (random) prime magic squares of order 12, 15, 17 and 22. Double Square: I build a double square of order 12. See how. Order 35: My program now generate a 35 order prime magic square. Order 124: Yes this high order was generate in semi-automatic mode in a time that I think below of order 35. The order 191 seems near my hand... Muncey Construction: I have suppose the steps used by Muncey to build his 12 order square. Other primes: Here the true prime sequence... All order: Here a riepilogative table for prime magic squares. Speed how fast are the programs?

45. Magic Square Solution(n=3,5,7,..) Demonstration for the magic square 5 x 5. (n=5, sum=65). 1st figure of 5x5; Put the first number in the middle column of the top row. http://user.chollian.net/~brainstm/odd.htm

The solution for the Demonstration for the Magic Square 5 x 5 (n=5, sum=65) Put the first number in the middle column of the top row. Put the next number in the box moved one column to the right and one row up. If the number exceeds a column or a row, place it in the opposite side of that column or row. Repeat step 2 'n' times just before you reach the original starting position. Place the next number in the same column one row below the last number and continue with step 2. Basically, you're placing the numbers in consecutive order diagonally up and to the right until all spaces are filled. Magic Square World Solution(n=4,8,..) Solution(n=6,10,..) Source Program ... Samples Shin, Kwon Young - brainstm@chollian.net

46. History Of Magic Squares magic squares have fascinated mankind throughout the ages, with examples being Chinese literature dating from as early as 2800 BC, when a magic square http://www.markfarrar.co.uk/msqhst01.htm

History Of Magic Squares Magic Squares have fascinated mankind throughout the ages, with examples being found in: Chinese literature dating from as early as 2800 B.C., when a Magic Square known as the "Loh-Shu", or "scroll of the river Loh" (see above), was invented by Fuh-Hi, the mythical founder of Chinese civilisation Greek writings dating from about 1300 B.C. the works of Theon of Smyrna in 130 A.D. use by Arabian astrologers in the ninth century when drawing up horoscopes Arabic literature, written by Abraham ben Ezra, dating from the eleventh century India, dating from the eleventh or twelfth century, where the earliest fourth order magic square was found, in Khajuraho the writings of the Greek mathematician, Emanuel Moschopulus, whose works now reside in the National Library in Paris more recently, magic squares appeared in Chinese literature during the latter part of the posterior Chou dynasty (951 - 1126 A.D.) or the beginning of the Southern Sung dynasty (1127 - 1333 A.D.) the works of Cornelius Agrippa, a German physician and theologian from the sixteenth century, who constructed seven magic squares, of orders three to nine inclusive, which he associated with the seven planets then known (including both the Sun and the Moon)

47. Magic Squares And Dihedral Group D4 Java applet generates odd ordered squares, with options to transformations by rotations and reflections. Souce code provided. http://www.wou.edu/~burtonl/magic.html

Magic Squares The sum of the numbers in each row, column and main diagonal is the same. It is called the magic constant . In the applet below, magic squares of order (size) 3, 5, 7 and 9 may be explored. A magic square of odd order n contains the integers from 1 to n , arranged so that the rows, columns and diagonals sum to the magic constant ( n n + 1)) / 2. The middle number in each square is ( n + 1)/2. Read more about the squares and the applet here applet-java must be enabled Applet Controls Reset : reset to initial square R0 : rotate degrees clockwise R1 : rotate 90 degrees clockwise R2 : rotate 180 degrees clockwise R3 : rotate 270 degrees clockwise FH : flip horizontally FV : flip vertically FX+ : flip on the line y = x FX- : flip on the line y = -x Order : select the size of the square The Applet The buttons in the applet apply a set of motions to the square which is known in group theory as the dihedral group D4 . The motions are all rotations and reflections of the square in the Euclidean plane, which are rigid in the sense that they do not bend, tear or stretch the square. These rotations and reflections preserve the magic quality of a magic square.

48. Notes On Magic Squares An introduction, including computer estimates of squares of different orders. http://www.trump.de/magic-squares/index.html

Notes on Magic Squares and Cubes by Walter Trump Table of contents Number of magic squares The results of historical and computer enumeration of magic squares All ultramagic squares of order 7 A complete description (10 pages) how all of these squares can be determined. The successful search for the smallest 'perfect' magic cube. How many magic lines can a magic cube of small order possess? Magic series (New!) Read all about these bricks of magic objects. Number of normal order-6 squares Estimate of the number of 6x6-squares (strategy and source code). Links to great sites about magic squares and cubes Each of these sites covers the subject very detailled. Walter Trump

49. 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. http://www.kanadas.com/ccm/magic-square/

Japanese version [English version] [Temporary Mirror Page (Fast!)] [Original Page (Newer?!)] Magic Squares 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

50. Magic Square Page I have collected some links to WWW pages on magic squares. Simple magic square checker and gif maker by Fabrizio Pivari; magic square Solver for PC http://www.kanadas.com/puzzles/magic-square.html

Magic Square Page I have collected some links to WWW pages on magic squares. This list includes a link to my page that contains a Java applet that solves magic squares stochastically. This page also contains a link to off-line information Indices Suzanne Alejandre: Magic Squares ``What is a Magic Square?'', History, Mathematics, Geography, ... Magic Squares (Database) by M. Suzuki, including a detailed reference list English Page Japanese Page Magic Square, Cube, Hypercube... Magic Square by Constraint Programming, etc. ``Magic Squares: Filling the Gaps'' Freudenthal-Tits Magic Square Methods of Solving Magic Squares Scalable Methods Solving Magic Squares by a stochastic method called CCM (Chemical Casting Model) a mirror by the author of this page Constraint Programming: Magic Squares: Filling the Gaps'' Mixed Integer Programming: Sci.Op-research 1994: MIP formulation of Magic Squares Magic Square: Java applet An Algorithm for odd magic squares Test Nonscalable Methods A Method for 6x6 Squares Magic Square Example Problems by Harvey Heinz Unusual Magic Squares, etc.

51. Nested Magic Squares Definition of nested squares. Generation of even and odd-sided squares, algorithms and matlab function. Samples of order up to 99 are presented. http://robotics.eecs.berkeley.edu/~srinath/magic/

Home Page Favorite Links Friends Research ... Personal Page N e s t e d M a g i c S q u a r e s Algorithm by: Samavedula Sita Rama Sastry Nested Magic Squares are magic squares with the special property that each sub-magic square within the original square is also magical, i.e the rows, columns, and diagonals add up to the same number (although to a number different than the original sum,... obviously) For example, in the 8 x 8 magic square: the inner 6 x 6 square is also magical and the magic sum is 195. This nested nature persists all the way to the inner 4 x 4 square. The page describes an algorithm to generate "nested magic squares" of even and odd sided dimensions. The algorithm was developed by Sri. S. S. R. Sastry. Although the algorithm for generating odd sided magic squares is much simpler, it was the algorithm for even sided magic squares which was developed first. For even sided magic squares, the methodology differs slightly for "double even sided magic squares" (i.e when dimension is divisible by 4) and "single even sided magic squares" (i.e when dimension is divisible by 2 but not by 4). Tomas Ullrich jr. (tom_ullrich AT hotmail DOT com) has an excellent implementation of this algorithm on in Excel. Download

52. Constructing All Magic Squares Of Order Three Mathematical approach to the generation of order three squares by applying MacMahon's partition analysis. http://people.brandeis.edu/~maxima/files/papers/magicsquare.pdf

53. Solving Magic Squares An attempt to solve for all possible magic squares, a mathematical treatment. http://www.mathpages.com/home/kmath295.htm

Solving Magic Squares Note 1 The Determinants of 4x4 Magic Squares Magic Square of Squares Orthomagic Square of Squares ... Return to MathPages Main Menu

54. Franklin's Magic Squares How Benjamin Franklin constructed his magic squares. Autobiographical extract included. http://www.mathpages.com/home/kmath155.htm

Franklin's Magic Squares Return to MathPages Main Menu

55. 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 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

56. Magic Square, Inc. L3 is a new highly interactive Internet graphics scripting technology. L3 is unique in its exceedingly small footprint and smooth animation. http://www.magicsquare.com/

Founded in 1996, Magic Square Inc. develops powerful Internet technologies, and educational software. is a highly interactive Internet graphics scripting technology. L3 is unique in its exceedingly small footprint and smooth animation. L3 is an excellent tool for both the p rofessional web designer, and for the amateur designer. L3 is also an ideal environment for teaching computer programming in middle-schools, high-schools and colleges. Available on Windows 95/98/NT. Visit the L3 site to learn more, and to download the free fully functional evaluation copy LogoMation is a low-cost, easy to learn computer programming environment, for middle and high schools. Available on Windows and Macintosh. Visit the LogoMation site to learn more, and to download your free evaluation copy.

57. Skeleton Web Demonstrates methods and algorithms for squares and cubes. http://web.onetel.com/~johnkim/

SKELETON MAGIC SQUARES (and cubes) This square of 25 numbers is said to be 'magic' because the sum of every row, every column and both diagonals is the same The program reveals how Magic Squares can be constructed on ANY NUMBER greater than 2 Discover the underlying structures of 'magic squares' and hence how to form such squares. Find out how a 4 x 4 square can be made 'super-magic' Extend the use of the `skeleton' method to form MAGIC CUBES If you are a mathematician perhaps you can . . . (1) demonstrate how many of all possible magic squares can be generated by this method. (2) find ways to make the cubes MORE magic? Fun for the layman - intriguing for the expert ENTER THE PROGRAM HERE Comments and queries to: john@ tullyj.freeserve.co.uk THE GOLDEN EGG A STORY FOR YOUNGSTERS Touring a Stately Home, Joey does not share his aunt's appeciation of a jewel-encrusted golden egg. He'd rather explore with the Earl's son. Their fun is Click here to obtain the ebook from: PLAYS FOR YOU TO STAGE The Miracle of Kofutu Getting Rid of Roddy Two full-length comedy dramas guaranteed to keep an audience hooked while packing a message for our times CLICK HERE TO VISIT THE SITE ONE ACT PLAYS Love From Michael King of Argos Highly original and very funny!

58. Panmagic Square -- From MathWorld Demonstration of the properties of a panmagic square. http://mathworld.wolfram.com/PanmagicSquare.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Recreational Mathematics Magic Figures Magic Squares ... Vantieghem Panmagic Square If all the diagonals including those obtained by "wrapping around" the edgesof a magic square sum to the same magic constant , the square is said to be a panmagic square (Kraitchik 1942, pp. 143 and 189-191). (Only the rows, columns, and main diagonals must sum to the same constant for the usual type of magic square.) The terms diabolic square (Gardner 1961, pp. 135-137; Hunter and Madachy 1975, p. 24; Madachy 1979, p. 87), pandiagonal square (Hunter and Madachy 1975, p. 24), and Nasik square (Madachy 1979, p. 87) are sometimes also used. No panmagic squares exist of order 3 or any order for an integer . The Siamese method for generating magic squares produces panmagic squares for orders with ordinary vector (2, 1) and break vector (1, -1).

59. Chinese Mathematics: Rebecca And Tommy It wasn t until the +13th century that the development of magic squares was introduced into the Some of the magic squares he made are very complicated. http://www.roma.unisa.edu.au/07305/magicsq.htm

Magic Squares Although originally of very little mathematical significance, magic squares were discovered by the Chinese emperor Yu the Great in approximately '2000 B.C' (Sanford, 1958, p74). As legend goes, Yu was presented two charts or diagrams by miraculous animals, during his reign of governing the empire. They were 'deemed to possess magical properties' (Rouse Ball, 1960, p119). 'The Ho Thu (diagram 8) was the gift of a dragon horse which came out of the Yellow River, and the Lo Shu (diagram 7) the gift of a turtle from the River Lo. The former (the 'River Diagram') was generally described as green, or in green writing, and the latter (the 'Lo River Writing') was traditionally red' (Needham, 1959, p56). Diagrams 7 and 8 respectively, the Lo Shu Diagram and the Ho Thu Diagram. The animals didn't simply hand the emperor these, Yu the Great actually found them physically on the creatures, for example the Lo Shu was on 'the back of the tortoise' (Sanford, 1958, p74). They were considered to be magical as it was recognised that the sum of the numbers in every row, in every column, and in each major diagonal was the same. Clearly the Lo Shu is a straight forward magic square in which the integers added up along any column, row or major diagonal add up to 15 (see diagram 9). The Ho Thu is arranged a little different. Discarding the centre two integers (5 and 10) both the odd and even sets add up to 20 (see diagram 10). Note in diagrams 9 and 10 originally even numbers (yin numbers) were represented in black and odd (yang) in white.

60. 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 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

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