21. Grogono Magic Squares Home Page Conversion To convert a magic square starting at one to a traditional square, one is added to each cell in the square. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

22. Amof:Info On Magic Squares The Amazing Mathematical Object Factory generator makes up to 10 squares of order 1 to 10 with just a few clicks. Brief history and introduction included. http://theory.cs.uvic.ca/~cos/amof/e_magiI.htm

Information on Magic Squares Description Example History Applications ... Links Description of the Problem Draw a 3 by 3 grid on a piece of paper. Now try to place the numbers 1 through 9 into the squares in such a way that all columns, all rows and both diagonals add up to the same amount. The result is known as a magic square In general, a magic square is an n x n matrix of numbers where each row, each column and both diagonals add up to n n The simplest magic square is the 1 by 1. This is simply a 1. There are no 2 by 2 magic squares, but there are for 3 by 3, 4 by 4, 5 by 5, and so on. Example For example, all the magic squares of size 3 by 3 are shown below: Notice that all of these squares can be obtained from the first one through flips and turns (rotations) of the first magic square. AMOF will only generate one of these 8 squares. A Brief History Magic Squares are claimed to go back as far as 2200 BC when the Chinese called them lo-shu. According to legend, the pattern was first revealed on the shell of a turtle that crawled out of the Lo River in the twenty-third century B.C. Here is the lo-shu magic square: The first indication of any mathematical investigation into magic squares was from Cornelius Agrippa. In the early part of the 15th century in Europe, he constructed magic squares from orders 3 to 9. He associated these squares with the planets then known, including the sun and moon.

23. Associative Magic Square -- From MathWorld An nxn magic square for which every pair of numbers symmetrically The numbers of associative magic squares of order 1, 2, are 1, 0, 1, 48, . http://mathworld.wolfram.com/AssociativeMagicSquare.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 Associative Magic Square An magic square for which every pair of numbers symmetrically opposite the center sum to . The Lo Shu is associative but not panmagic . The numbers of associative magic squares of order 1, 2, ... are 1, 0, 1, 48, ... (Sloane's Order four squares can be panmagic or associative, but not both. Order five squares are the smallest which can be both associative and panmagic , and 16 distinct associative panmagic squares exist, one of which is illustrated above (Gardner 1988). The numbers of associative panmagic squares of order 1, 2, ... are therefore 1, 0, 0, 0, 16, ... (Sloane's SEE ALSO: Magic Square Panmagic Square [Pages Linking Here] REFERENCES: Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments.

24. Allmath.com Features free flashcards, a magic square game, biographies of mathematicians, and other resources. http://www.allmath.com/

Online Dictionary allmath.com Allmath Homepage Math Tools Flash Cards!!! Metric Converter Games The Magic Square Reference Pages Biographies of Math Math Glossary Multiplication Table Metric Conv. Factors ... Other Math Links Ask The Experts Visit Dr. Math Visit our Dictionary! ALLwords.com About allmath.com Welcome! Privacy Statement Who are We? Advertise Other Great Math Links! Fun And Games Kid References Kid Resources Math Articles ... Allsites LLC allmath.com, mathbook.com and "virtual mathbook" ® are all service marks of Allsites LLC Periodic Table Metric Conversion Math Book Store ... Mortgage Calculator

25. Shin's Magic Square World The Perfect Solution For the MAGIC SQUARE . .. .KOREAN( ). .. . Since July 1997 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

26. Magic Squares , properties and online construction. Tour of famous squares. (English/German) Flash optional...... http://www.magic-squares.de/magic.html

url = new Array( 26,0,0, 0,"general/squares/squares.html","Information about magic squares", 0,"general/squares/information.html","Definition of magic squares", 0,"general/squares/magicsum.html","Magic sum", 0,"general/squares/transformations.html","Reflections and rotations of magic squares", 0,"general/squares/count.html","Counting magic squares", 0,"general/squares/number.html","Number of magic squares", 21,0,0, 0,"general/additional/links.html","Links to web sites with magic squares", 0,"general/additional/sitemap.html","Sitemap of www.magic-squares.de" ); The famous Loh-Shu from 2800 B.C. It is the oldest known magic square of the world. And you have the possibility to create millions of other, different magic squares online on these pages. Comments, Remarks and Questions to Holger Danielsson Letzte Änderung: 26.06.2003

27. Education 4 Kids Sponsored By "your Name Here" (a Free Educational Note This was called "magic square", but the name has been changed since it was not really a "magic square" where all rows, columns and http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

28. Math Forum Suzanne Alejandre Magic Squares What is a magic square? Allan Adler defines and discusses some special properties of magic squares. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

29. Math Forum Allan Adler - What Is A Magic Square? What is a magic square? http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

30. Magic Squares, Stars, And Other Pages on magic squares, stars (original material) and miscellaneous number patterns. Downloads include BASIC programs, Excel and .zip files. http://www.geocities.com/~harveyh/

M I hope you enjoy these examples from my collection of number patterns. This site has three sections, with pages on magic squares, magic stars (a lot of original material) and miscellaneous number patterns. This site should be of interest to middle and high school students and teachers, and anyone interested in recreational mathematics. Wherever there is number, there is beauty. Proclus (410-485 A.D.) Star Updates A new page containing additions to the magic stars site. It also contains links to three new pages of material from Simon Whitechapel, Jon Wharf, and Andrew Howroyd. All three have contributed significantly to magic star knowledge, including confirming total solution counts. Cube Site Map Feb. 22, 2005, I added links to the 41 pages of my cube site to this site map. Also, I corrected some links and minor cosmetic changes to a number of pages. Search Engine Dec. 17, 2004. I installed a search engine service for help in locating objects on my Geocities and Shaw Web sites. The index for both sites is combined, so entering a search term will supply a combined list of hits from both sites. Magic Cubes site In Sept. 2003, I added another button to the top of each page in this site. It points to a new site of 30 + pages discussing the history and many features of magic cubes.

31. Puzzling Phenomenon Sweeps British Papers feng shui qualities of the "magic square " in which the sum of every row or column of numbers organized in, say, a threeby-three square, is http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

32. Magic Squares Examples of many different types of magic squares. Other sections on orders 5 to 12 magic stars and number patterns. Recreational mathematics links. http://www.geocities.com/~harveyh/magicsquare.htm

M agic Squares I hope you enjoy these examples of a variety of magic squares. This section of my site consists mostly of examples, with a minimum of explanation and theory. Refer to the other two sections for Magic Stars and Number Patterns. This site should be of interest to middle and high school students and teachers, and anyone interested in recreational mathematics. Order-3 # 1 of 1 basic solution. Order-4 # 290 of 880 basic solutions. Order-5 # 1233 of 3600 pandiagonal solutions Magic Squares are a form of number pattern that has been around for thousands of years. For a pure or normal magic square, all rows, columns, and the two main diagonals must sum to the same value and the numbers used must be consecutive from 1 to n , where n is the order of the square. Many variations exist that contain numerous other features. I show on these pages samples of the large variety of magic squares. My descussions will be limited to brief comments on the individual illustrations. Perhaps in the future, I will add more in depth information in the way of history, theory, construction methods, etc. Acknowledgments : As with all the mateial on this site, most of these illustrations are original with myself or I consider them in the public domain (i.e. I have multiple sources for the illustration). Many of the more unusual figures are one of a kind and I so acknowledge the author with thanks for permission to use them.

33. Magic Square From MathWorld A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, , n^2 arranged such that the sum of the n numbers http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

34. Allmath.com - Math Site For Kids! Home Of Flashcards, Math Features free flashcards, a magic square game, biographies of mathematicians, and other resources. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

35. Magic Squares And Hyper Cubes Includes Java Applet and source code of square and hyper cube generator. Options for dimensions and offset. http://net.indra.com/~charliek/

Introduction Magic squares are arrangements of numbers into squares, cubes, or hyper-cubes where the sum of each row and each column is identical. Additionally, the sum along the "principal diagonals" is also equal to the same number. The magic square below was known in antiquity in China, Greece, and Egypt. It is interesting to note that subtracting a constant from each cell in a magic square or hyper-cube does not change its essential property: namely, that the sum of any row, column, or principal diagonal is identical. For example, the following magic square is obtained by subtracting 5 from each cell in the magic square above: An interesting property about this square is that all rows, columns, and principal diagonals sum to 0. Also, it is easy to visually recognize symmetry in the square. This web site contains: Examples Instructions Applications Theory ... Symmetry - examples using "tuples" Generator - program for generating magic squares and Hyper Cubes Other Web Sites for Magic Squares and Cubes Please see the Instructions Page before viewing the Magic Squares and Hyper Cubes Generator Site Map: Postal address P.O. Box 567; Niwot, CO 80544

36. More Magic Squares Look at more 4 x 4 magic squares to see if they are panmagic squares. http://www.mathcats.com/explore/puzzles/magicsquare4x4.html

math cats explore go back Magic Squares Click on any magic square below and travel to a magic carpet filled with that square. Use the movable picture frame on each page and place it around ANY numbers. Do you get another magic square? Do all the groups of cells shown in green above still add up to the same number? Keep moving the frame around and see how many different magic squares you can find on each page!

37. Puzzle 297. Queens On Magic Squares Try to put 8 queens on an 8x8 square such that the queens sit on prime numbers. Readers' solutions provided. http://www.primepuzzles.net/puzzles/puzz_297.htm

Puzzles Puzzle 297. Queens on magic squares Dear friends: I will be traveling for the next 15 days through China. If I can get a stable contact with the server of my site from my laptop, I will try to continue posing puzzles, every Saturday morning. Otherwise, the next puzzle will come the next 15/1/2005. From the bottom of my heart, enjoy this season holydays and receive my very best wishes for the next year, 2005. Thanks for your continuous support participating in this special circle of friends of the prime numbers. Anurag Sahay poses the following puzzle: Construct a magic square n x n (using the numbers 1 to n ) and place n queens only on these cells which contain prime numbers, such that no queen can take any other queen. 1. What is the smallest magic square (n) having solution? 3. Redo the exercises 1 & 2 with one additional condition: " one of the diagonals should also contain prime numbers only " Sahay Queens on bold prime numbers: 23,19,17,53,5,47,31,43.

38. Grogono Magic Squares Home Page magic squares home page a website for maths lovers and recreational mathematicians. http://www.grogono.com/magic/

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. 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 Recent Addition Do it yourself! 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. 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.

39. Make Your Own 5x5 Magic Square Make your own magic squares automatically generate a 5x5 magic square. http://www.grogono.com/magic/makeyourown5.php

Home Page Make Your Own Choose by Size How Many? ... Grogono Home Size: Index Make Your Own: Any Prime Make Your Own Magic Square: 5x5 The magic square below is the sum of the patterns that these numbers make. Use them or enter new ones and then click To understand the structure, move the mouse over the blue arrows and wait. Look at the pattern of your numbers in the square below The Bottom Right number (under the blue down arrow) above is the first number in the sequence for the square. By convention it is "1", but any number can be used. Try changing the bottom row above: use "5,4,3,2,1,0" instead, do you get the same result as "4,3,2,1,0,1"? You can also try experimenting by writing in your own numbers in the red squares. To see the totals, press The numbers at the top and the left are the sums for the diagonal rows - including the broken diagonals. Explanation: The numbers in the Red Squares form the 3x3 magic Square. The numbers beside the Red Squares show the totals for each row. The horizontal and vertical totals are to the right and below in green squares. The other, blue, squares show the diagonal totals - including all of the "broken diagonals". You can make your own Magic Square in two ways. Try both methods:

40. Magic Square Take the addition challenge and play the magic square. The objective of the game is to place a number into each square so that the sum of the squares equal http://www.gomath.com/htdocs/game/magic/

Take the addition challenge and play the Magic Square. The objective of the game is to place a number into each square so that the sum of the squares equal to the number on the bottom and to the right of the square. The sum of the numbers you entered must equal to the number on the bottom and to the right of the square when added vertically, horizontally and diagonally. Click on New Game to start playing. Easy Medium Hard Back to GoMath

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