21. Magic Squares 4 9 2 3 5 7 8 1 6 This is a magic square of order 3 (three numbers to the sideof the square). Does this passage have to do with magic square numerology? http://www.netmastersinc.com/secrets/magic_squares.htm

Mark Swaney on the History of Magic Squares This is a magic square of order 3 (three numbers to the side of the square). If you add up any row, column, or diagonal, it sums to the same number, 15. There are magic squares of order 4, 5, 6, etc. See this link for a listing of magic squares of order 3 through 11: http://www.pse.che.tohoku.ac.jp/~msuzuki/MagicSquare.byMATLAB.html My friend Mark Swaney has been working on the history of Magic Squares and has said yes to my passing on some of his preliminary results with the following warning: You gotta tell them that it's just ripped hot off the neurons, and may have a detail or two out of place. I'm reading all this stuff and then roaring off an epistle. Later, I always think I should have done it differently, but what the hell? Also, I find that I like to write a lot of text when I'm feeling radioactive. After Mark's review of the history of Magic Squares, I have listed some further information and references on the squares. Mark: I am still getting references and picking up information, so the following is subject to revision and

22. Magic Square Design Describes how to create several different types of magic squares using patterns. http://geocities.com/bcenrichment/enrichment/math/woodmath.html

23. Shin S Magic Square World What I m saying here is It s Not Impossible!! . Solutions for the 3 types ofmagic square. Other magic square links. Mutsumi Suzuki s magic square Page http://user.chollian.net/~brainstm/MagicSquare.htm

24. Book For Sale magic square Lexicon Illustrated By Harvey D. Heinz and John R. Hendricks ISBN09687985-0-0, 228 pages 5 ½ x 8 ½, perfect bound, laminated cover. http://www.geocities.com/~harveyh/BookSale.htm

B ook for Sale Introducing a new reference book on magic squares, cubes, tesseracts, magic stars, etc. Magic Square Lexicon: Illustrated By Harvey D. Heinz and John R. Hendricks ISBN 0-9687985-0-0, 228 pages 5 ½ x 8 ½, perfect bound, laminated cover. 171 captioned illustrations and tables, 239 terms defined, 2 appendices of bibliographies. This book defines 239 terms associated with magic squares, cubes, tesseracts, stars, etc. Many of these terms have been in use hundreds of years while some were coined in the last several years. While meant as a reference book, it should be ideal for casual browsing with its almost 200 illustrations and tables, 171 of which are captioned. While this book is not meant as a "how-to-do" book, it should be a source of inspiration for anyone interested in this fascinating subject. Many tables compare characteristics between orders or dimensions. The illustrations were chosen, where possible, to demonstrate additional features besides the particular definition. (from the back cover) An example entry Summations The magic sum for an n-Dimensional Magic Hypercube of Order m is given by: S = m(1 + m n In a magic object, there are many lines that produce the magic sum. The table below, shows the minimum requirement of the number of lines for various types of magic hypercubes and is derived from the following equation:

25. Magic Squares Home Page Dedication. The magic square section of this website is dedicatedto my father EB Grogono (1909 1999). What is a magic square? http://www.grogono.com/magic/index.php

Home Page Make Your Own Choose by Size How Many? ... Grogono Home Size: Index Magic Squares Home Page Introduction. A Magic square is intriguing; its complexity challenges the mind. 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 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. Dedication. This Magic Square website is dedicated to my father E.B. Grogono (1909 - 1999) and was originally created at his bedside during his last illness. My fondest memories of him, from my earliest childhood to the final days of his life, center on his ability to transmit his love for, and fascination with, mathematics and science. Revision This revision uses up to date technology to make the website easier to manage and the material has been re-arranged to make it more accessible. A glossary has been added and the index system has been revised.

26. Magic Squares Home Page The order 5 square is the smallest odd number panmagic square which canbe made using a formula. Either choice produces a pan-magic square. http://www.grogono.com/magic/formula.php

Home Page Make Your Own Choose by Size How Many? ... Grogono Home Size: Index The Formula for Magic Squares Why this "Formula" Page. One of the commonest questions I receive by e-mail is: "What is the formula for Magic Squares?" Rather than go on answering each e-mail individually, I thought I would write this page. So, as promised in the introduction page, here is "The Formula". A Single Formula? Unfortunately, No! . It would be nice to find that a single simple equation would generate magic squares of any order. With the diversity of patterns used to make the different squares, it is obvious that no single equation could exist. In addition, some of the equations must be fairly complex. There is, however, a simple formula for the prime number regular pan-magic squares, e.g., for orders 5, 7, 11, 13, etc. Knight's Move Basis. The following formula is based on the Knight's Move technique for making magic squares. It is broken into two components, one for generating the low order bit pattern and one for the high order bit pattern. MOD(x - 2y, 5)

27. Magic Squares Applet Java source files. 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 questions/suggestions, mailto: suzanne@mathforum.org

28. 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

29. Magic Square Page 6, 3, 9. magic square Page. I have collected some links to WWW pages on magic squares. NonscalableMethods. A Method for 6x6 Squares. magic square Example Problems. 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.

30. Magic Squares With Odd Rows And Columns Describes a simple method to create magic squares with an odd number of rows and columns. A workthrough of 3x3 square. http://www.betnorth.com/magicsquares.htm

MAGIC SQUARES AND HOW TO MAKE SOME Amaze your friends and educate your children A magic square is a square of integers where all the rows and columns plus the diagonal add up to the same number. This page describes a simple method of forming a magic square with odd rows and columns. The method is a surprisingly simple one and mainly consists of writing consecutive numbers down and to the right. Lets write down the rules and then we will create a 3 by 3 magic square which should explain the rules as we go along. 1.Find the middle square and put the number "1" in the square below 2.Put the next number down 1 row and across 1 column. I.e. diagonally down. This is the general rule from now on. Just go down diagonally with consecutive numbers. 3.If there is no row below , put the number in the first row, next column to the right.I.e. "pretend" the 1st row is the one after the last. Similarly, if there is no column to the right, put the next number in the first column, one row down. 4.A special case of the above is where you are in a corner and there is no row down AND no column to the right. In this case go down 2 rows, using the "pretend" about columns and rows mentioned above. This rule also applies where there is already a number in the square you want to go to, which is a bit like an internal corner.

31. The Zen Of Magic Squares, Circles, And Stars: Ronald, 93 Bruno, Giordano, 3334 Buddhism, xiii-xv, 28-30 Butterflies, xi-xii, xxCameron cube, 120 Carus, Paul, xix, 204-209 Categories, magic square, 37 Chhi 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

32. All You Wanted To Know About Magic Squares magic squares. I hope to show you in these pages some of the many aspectsof magic squares, and maybe kindle that spark in you too. http://www.markfarrar.co.uk/msfmsq01.htm

Magic Squares "I have often admired the mystical way of Pythagoras, and the secret magic of numbers" - Sir Thomas Browne, 1605 - 1682 The above quotation accurately describes my own fascination with numbers generally, but also with Magic Squares (or Quadramagicology , as " New Scientist " magazine recently called it). I hope to show you in these pages some of the many aspects of Magic Squares, and maybe kindle that spark in you too. The information shown on the pages indicated below is mainly taken from my book on Magic Squares, which I wrote as my "entrance examination" to become a Member of The Magic Circle , although other sources have also been used: What Is A Magic Square? History Of Magic Squares Magic Totals Odd Magic Squares ... Online Store Subscribe to magicsquares Powered by groups.yahoo.com Click on this house to return to my Home Page Created: Sunday 28th December, 1997 Contact Me (since 22 May 1999)

33. Franklin's Magic Squares Covers how Franklin constructed his magic squares including a autobiographical extract. http://mathpages.com/home/kmath155.htm

Franklin's Magic Squares Return to MathPages Main Menu

34. History Of Magic Squares Chinese literature dating from as early as 2800 BC, when a magic square known asthe LohShu , or scroll of the river Loh (see above), was invented by Fuh-Hi 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)

35. Skeleton Web Demonstrates methods and algorithms on how to make magic squares and cubes. http://web.onetel.net.uk/~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 ALL MAGIC SQUARES can be constructed Discover the underlying structures of 'magic squares' and hence how to form a magic square based on ANY number from 3 upward 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 serious student of mathematics perhaps you can . . . d emonstrate that ALL po s sible magic squares are ultimately based on the same underlying structures (2) find ways to make the cubes MORE mag ic? Fun for the layman - intriguing for the expert ENTER THE PROGRAM HERE C omments and queries to: john@ tullyj.freeserve.co.uk Force of Gravity? There is no such thing! A program which argues that other forces in the universe make the concept of a `Force of Gravity' superfluous. Only elementary mathematics is used. To visit the site click here

36. Magic Squares Applet magic squares Applet. Having trouble running this applet? Downloadthe lastest version of the Java Run Time Environment. Click the http://www.mste.uiuc.edu/java/michael/magicsquares/

Magic Squares Applet Having trouble running this applet? Download the lastest version of the Java Run Time Environment Click the 'Get It Now' link in the top right corner of the page. The Source Files.... MagPuz.java MagCan.java MagControls.java ColorsWindow.java Credits: 1996: Mike Morton wrote this applet Pavel Safronov updated and recompiled this applet. Michael McKelvey updated and recompiled this applet. Please Note: This applet, which is an update of the original Magic Squares Applet , has been having problems running on the following system configurations: Windows 98 with Internet Explorer 6.0 We have, however, seen it working on the following configurations: Mac with both Safari and IE 5.2.3 Windows 98 with IE Windows Me with IE 6.0 (Java Plug-in 1.4.1) Windows Me with NN 6.1 (Java Plug-in 1.4.1) Windows 2000 with IE Windows XP with IE 6.0 (Java Plug-in 1.4.2) An interesting property discovered by some users is that when the updated version does not run, the original will, and vice versa. If you have any questions/suggestions, send an email to:

37. Ivars Peterson's MathLand: More Than Magic Squares Typically, a magic square consists of a set of integers arranged in the form ofa square so that the sum of the numbers in each row, each column, and each http://www.maa.org/mathland/mathland_10_14.html

Search MAA Online MAA Home Ivars Peterson's MathLand October 14, 1996 More than Magic Squares "In my younger days, having once some leisure (which I still think I might have employed more usefully), I had amused myself in making . . . magic squares." Benjamin Franklin, who made this comment in a letter written more than 200 years ago, was certainly not the first to experience the fascination of magic squares. People have been toying with these number patterns for more than 2,000 years. Typically, a magic square consists of a set of integers arranged in the form of a square so that the sum of the numbers in each row, each column, and each diagonal add up to the same total. If the integers are consecutive numbers from 1 to n ^2, the square is said to be of n th order. Here's an example of a magic square of the fourth order, made up of the first 16 integers. The sum of the numbers in each row, column, and diagonal is 34. There are 880 possible magic squares of the fourth order, not counting reflections or rotations of each pattern. One of the most remarkable of these squares is one that dates back to India in the eleventh or twelfth century. Notice that not only the rows, columns, and diagonals add up to 34 but also the corner 2 x 2 subsquares. And there's more! The four corner numbers add up to 34, as do the four numbers in the center. Other subsquares (such as 3 + 10 + 6 + 15) give the same result. It's also possible to find "split" subsquares and "split" diagonals that work: 7 + 2 + 14 + 11, and so on. In fact, there is an astonishing number of different ways to get the sum 34 out of this particular magic square.

38. Math Forum: Allan Adler - Multiplying Magic Squares Includes pages, most for use in the classroom, covering what a magic square is, the math behind them and how they are constructed. http://mathforum.org/alejandre/magic.square/adler/

A Math Forum Web Unit Allan Adler's Multiplying Magic Squares Suzanne Alejandre's Magic Squares About Allan Adler Classroom activities How to construct magic squares ... Contact Us http://mathforum.org/ Web page design and graphics by Sarah Seastone

39. Cut The Knot! In Merlin s magic square, an article that appeared in The American Mathematical Monthlyin 1987, Don Pelletier explored the mathematical apparatus behind a toy http://www.maa.org/editorial/knot/Merlin.html

Search MAA Online MAA Home Cut The Knot! An interactive column using Java applets by Alex Bogomolny Merlin's Magic Squares April 1998 In Merlin's Magic Square , an article that appeared in The American Mathematical Monthly in 1987, Don Pelletier explored the mathematical apparatus behind a toy game known as MERLIN . The game is not quite trivial and the mathematics is simple enough to provide an entertaining exercise in a Linear Algebra class. The original game is played on a 3x3 array of buttons that toggle between two states. The goal of the game is to achieve a target configuration of button states by pressing the buttons. The difficulty lies in that pressing a button alters its state but also toggles states of some neighboring buttons. The applet below generalizes the game in three ways: Buttons are multistate, The target configuration is modifiable, The manner in which pressing a button affects its neighbors is selectable. The applet consists of two 3x3 arrays. On the left, the small one shows the target configuration. To modify the target configuration, click on the squares you want modified. On the right, a bigger one holds the puzzle itself and, if the Hint box is checked, the hint or, rather, the solution to the puzzle. The hint configuration is also modifiable and the current state of the puzzle changes accordingly. States are represented by a cyclic arrangement of digits - the residues modulo the number of states. I allow only 2, 3, and 4 state buttons for two reasons. For one, with more states the puzzle grew too difficult for me to solve. The second reason will become apparent from the puzzle's theory.

40. Multimagic Squares magic squares which remain magic after their entries are raised to various powers. Examples, constructions, bibliography and links compiled by Christian Boyer. Site in English and French. http://www.multimagie.com/

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