81. Fractional Magic Squares It took her quite a while because she didn’t know that the sum of a magic squarewas always three times the number in the centre. Call this magic square C. http://www.nzmaths.co.nz/PS/L4/Number/FractionalMagicSquares.htm

F ractional M agic Squares Number, Level 4 The Problem Tui has really begun to get the idea of magic squares. She decided to make all of the magic squares that she could using the fractions 7/6, 4/3 3/2. How many can she make? It took her quite a while because she didn’t know that the sum of a magic square was always three times the number in the centre. What is the problem about? First of all, if the class hasn’t heard of magic squares, then you may need to tell them that a magic square is an arrangement like the one below where the vertical, horizontal and diagonal lines of numbers all add up to the same value. This ‘same value’ is called the sum of the magic square. Magic squares are interesting objects in both mathematics proper and in recreational mathematics. So they are objects that children should have heard about and experienced. The problems in this sequence give students the opportunity to use the new numerical or algebraic concepts that they will have acquired at that Level, along with magic squares. It’s a critical part of this and some earlier problems that three times the centre square is equal to the sum of the magic square. We prove this in the Extension to

82. CoolJava -- Magic Squares Click Here Home Normal Games magic squares. The object is to getall the numbers in order. The 0 square represents a blank spot. http://www.javacommerce.com/cooljava/games-normal/magicsquares.html

Home Normal Games > Magic Squares The object is to get all the numbers in order. The "0" square represents a blank spot. Click on a number next to that "0" square to move that number to the "0's" location. Good luck. Put the numbers in order so that they read 1-8. The is the 'empty' place. Click on any number next to and they will switch places. # of moves: How to install this script: Choice 1: You can view the source of this page by clicking the button and copying the source code that way. (experts only) Choice 2: Copy and paste the information below into your web document. Highlight the script, then press Ctrl-C to copy, and the Ctrl-V to paste within your document editor. Choice 3: View a basic HyperText Markup Language file with nothing but the script, and save the contents to your hard drive. Web Directory Home Corba FAQ Java Links ... Cetus Links permission from JavaCommerce Javacommerce is not sponsored by or affiliated with Sun Microsystems, Inc. You are free to copy these script and applets and use them on your site as long as the author information and credits is retained.

83. Magic Squares The 0 repesents the empty spot. Click a square next to the 0 to make themtrade places! Get the squares back in order and you WIN! magic squares. http://www.cybergrace.com/html/magic_squares.html

Here is a fun JavaScript game. Try to put the squares back in order. The repesents the empty spot. Click a square next to the to make them trade places! Get the squares back in order and you WIN! Magic Squares Put the numbers in order so that they read 1-8. The is the 'empty' place. Click on any number next to and they will switch places. # of moves: Home Knowing God Opportunities Advertise ... Add Your Site

84. Magic Square The 0 represents the empty spot, and click a square next to the 0 to make themtrade places! Get the squares back in order and you win! magic squares. http://www.uhealthy.com/english/funzone/game-magic-sq.htm

Lights Out! Magic Squares Blackjack Try to put the squares back in order. The represents the empty spot, and click a square next to the to make them trade places! Get the squares back in order and you win! Magic Squares Put the numbers in order so that they read 1-8. The is the 'empty' place. Click on any number next to and they will switch places. # of moves: Member Sign In:

85. MAGIC SQUARE VOCABULARY magic square VOCABULARY (Lois Hoshijo). Descriptor I chose to use a 3 x3 magic square that was an example from Art Scholastic (1993). The http://linus.icoe.org/reading/cal/magicsquare.htm

MAGIC SQUARE VOCABULARY (Lois Hoshijo) Descriptor and Setting: Content Area Literacy "Magic Square" Math Vocabulary Strategy—Lois Hoshijo used the "Magic Square" vocabulary strategy in her middle school classroom to assess students' knowledge of definitions of mathematical terms. The procedure that Lois used is described by her as follows: I chose to use a 3 x 3 Magic Square that was an example from Art Scholastic (1993). The numerical total for each vertical, horizontal, and diagonal row was 18. I organized my activity with three separate sections: a) the 3 x 3 square each labeled with a letter, b) the list of content area terms, and c) the list of definitions for those mathematical terms. Students were directed to match each term with the appropriate definition, while they also considered the numbers denoting the terms as well as the letters denoting the definitions. Once the match was made, the number of the correct mathematical term was placed in the proper space of the square that was marked by the letter of the matching definition. Students were not informed about what the correct total of the Magic Square was. If their matches were correct, the Magic Square would be completed.

86. Magic Squares magic squares. Quickie Introduction. A curious arrangement of numbers includeswhat is referred to as a “magic square”. This is a magic square of rank 3. http://www.halexandria.org/dward090.htm

Magic Squares Quickie Introduction A curious arrangement of numbers includes what is referred to as a “magic square”. The magic derives from the fact that numbers arranged in a square of equal sides all add to the same total, coming and going, up and down, and oft times even from an angle (diagonal). For example: Note that the total always adds to 15 (row, column or diagonal), the diagonals no longer necessarily add properly if either the row and/or columns are mixed, and the total of any three rows or columns is 45. This is a magic square of rank 3. One can also do a 4x4 magic square, e.g. Here the rows and columns add to 34, but in this particular case the diagonals do not. The 3x3 example above is considered Panmagic Diabolical, Nasik, or Pandiagonal , while the 4x4 above is merely magic. It is also possible to start with zero, instead of one, so that a possible 5x5 magic square is: http://www.grogono.com Included is a very brief, traditional history, which notes that “all magic squares have at least eight variations: the square can be rotated into four positions and each of these rotations can be reflected - for a total of eight variations of any one unique design. Most magic squares do not remain magic if one border is moved to the opposite edge - the change leaves the main diagonal no longer magic.

87. Magic Squares 1 A magic square is of order N is an NxN matrix containing the integers from1 toN 2 arranged so that the sum of each row, column and the cornerto-corner http://www.delphiforfun.org/Programs/magic_squares1.htm

Home Introduction About Delphi Feedback ... Site Search (bottom of page) Available Now (Click a duck below to expand program list if it is collapsed - MS Internet Explorer only) There is also a computer generated program index page here Beginners 10 Easy Pieces Bouncing Ball A Card Trick? Cards #1 ... Multiple of 12? Intermediate 15 Puzzle #1 The 9321 Problem Aliquot Sums (Friendly If not Perfect) Alphametics ... Word Ladder Advanced 15 Puzzle #2 Airport Simulation Akerue Astronomy Demo ... WordStuff 2 Contact Feedback Send an e-mail with your comments about this program (or anything else). Search WWW Search delphiforfun.org Problem Description A Magic Square is of order N is an NxN matrix containing the integers from1 to N arranged so that the sum of each row, column and the corner-to-corner diagonal are all equal. Write a program to create Magic Squares of odd order (1,3,5,7, etc.) up to order 51. For small squares we could use the brute force approach as used in the 3x3 matrix in the rotating sums problem. But recall from earlier exercises that there are X! ways to arrange X numbers and 10! is approaching the upper limit that can be solved quickly (10! is about 3.6 million). This implies that finding the solution by brute force for an order 5 square would require trying up to 25! permutations of the numbers from 1 to 25. Clearly not a practical approach. Like many puzzles of a mathematical bent, Magic Squares have been studied for hundreds of years. The algorithm used here for odd order Magic Squares was discovered at least 500 years ago. I'll leave it to the reader to search out the pertinent history.

88. Durer's Magic Square . Dürer s magic square is contained in a famous copper engraving, Melancholia , created in 1514 by German artist Albrecht Dürer.......Problem http://www.delphiforfun.org/Programs/durersSquare.htm

Home Introduction About Delphi Feedback ... Site Search (bottom of page) Available Now (Click a duck below to expand program list if it is collapsed - MS Internet Explorer only) There is also a computer generated program index page here Beginners 10 Easy Pieces Bouncing Ball A Card Trick? Cards #1 ... Multiple of 12? Intermediate 15 Puzzle #1 The 9321 Problem Aliquot Sums (Friendly If not Perfect) Alphametics ... Word Ladder Advanced 15 Puzzle #2 Airport Simulation Akerue Astronomy Demo ... WordStuff 2 Contact Feedback Send an e-mail with your comments about this program (or anything else). Search WWW Search delphiforfun.org Problem Description Dürer's Magic Square is contained in a famous copper engraving, "Melancholia", created in 1514 by German artist Albrecht Dürer. There are 86 different combinations of four numbers from the square that sum to it's magic number, 34! How many can you find? Albrecht Dürer is generally considered to be Germany's most famous Renaissance artist. He was about 20 years younger than Leonardo da Vinci and around 1500 became greatly interested in the relationship between mathematics and art. Leonardo and his contemporary, mathematician Pacioli, almost certainly influenced Dürer in these studies. In 1514 he created the copperplate engraving named "Melancholia I" which contained this magic square - the first magic square published in Europe. (Notice that the creation date of the picture, 1514, is contained in the bottom row of the square.)

89. The SATOR Magic Square The SATOR magic square. A magic square Chinese. On the shell of the tortoiseit was drawn a magic square of order three. magic squares http://www.math.unifi.it/~caressa/math/sator.html

The SATOR Magic Square A magic square is a square table (mathematicians call them "matrices") of numbers with the remarkable property that the sum of numbers on diagonals, rows and columns give the same number: of course one requires that the numbers appearing in the table are all different between them. For example is such that the sum of rows, columns and diagonal gives 15. This is a rather symmetric magic square since: the sum of numbers along sub-rectangles with six numbers is always 30 (this is obvious since the sum of rows and columns gives 15), and the sum of sub-triangles with four numbers is always 20 (also this is obvious). You may find a lot of links and books on magic squares: here I confine myself to a brief discussion just to introduce an unaware reader. The first appearance of a magic square in history can be traced back to XXIII century b.C: the Chinese emperor Yu (of the Hsia dynasty) was contemplating the Yellow River when he saw, emerging from its waters, a tortoise, a sacred animal to ancient Chinese. On the shell of the tortoise it was drawn a magic square of order three. Magic squares were considered, also due to this mysterious origin, as talismans and amulets which can cure the health both of body and spirit. From China their cult (and also recipes to build them) penetrated in India: for example in 1356 Narayana described several ways to construct magic squares of different orders.

90. Magic Square: Java Applet magic square Java applet. The applet below can is generated. Although in most cases only a semimagic square appears. Clicking the http://www.ouh.nl/open/eyn/applets/magicsq.htm

Magic Square: Java applet The applet below can be used to generate magic squares with odd sides (1, 3, 5 of 7). After some experiments you will probably discover the construction method, enabling you to make bigger squares for yourself. Here, a magic square is considered to be a sqare filled with integer numbers from a monotonic sequence, i.e. a sequence having a constant difference between consecutive elements, in such a way that all sums of rows, columns and both diagonals are equal. Instructions for using the program Choose the desired size of the square in the choice list in the upper left corner. If everything works properly a square of empty text fields appears, with comments on the sides and some extra text fields for row, column and diagonal totals. Type the horizontal and vertical positions of the first number of the abovementioned monotonic sequence in both text fields below the text 'Coordinaten beginvak:'. Next type the values of this first number and the constant difference to the right of the texts 'Basis' and 'Increment'. Then, clicking the button 'Maak vierkant' ( make square ), the desired square is generated. Although... in most cases only a

91. TOKYO MAGIC SQUARE ?TOKYO magic square?. CLAMPONLY WEB. lust date up2002/5/6. http://www.geocities.co.jp/Playtown-Dice/1629/clamp-e.html

lust date up:2002/5/6 ¡Ö¥«¡¼¥É¥­¥ã¥×¥¿¡¼¤µ¤¯¤é¡×¡ÖANGELIC LAYER¡× ¡Ö¤Á¤ç¤Ó¤¥Ä¡×¥á¥¤¥óCLAMPºîÉÊONLYƱ¿Í»ï¨Çä²ñ CLAMP CARNIVAL5 ²ñ¾ì¡§à ÔΩ»º¶ÈËÇ°×¥»¥ó¥¿¡¼£µF ¡ÖCLAMP CARNIVAL4¡×

92. 4cheaters - 2D Magic Square - Cheats, Tipps, Tricks, Codes Und Trainer Translate this page 4cheaters » PC » 2D magic square (64 Hits) 2D magic square. Trainer. Game-Info.2D magic square System PC Genre Denkspiel. Funktionen. Forum. Cheats drucken. http://www.4cheaters.de/pc/cheats/2d_magic_square_11560

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93. Using T-SQL With Magic Squares Learning how to use TSQL to populate magic square can sharpen your programmingskills for more practical tasks. Figure 1 is an example of a magic square. http://www.winnetmag.com/SQLServer/Article/ArticleID/9020/9020.html

Home Hot Topics Active Directory Migration ... My Account Within: SQL Server Magazine Entire Network Advanced Search LOG ON Driving the Enterprise Network NT XP The Smart Guide to Building World-Class Applications Backup and Restore Change and Configuration Management Data Modeling Data Transformation Services ... August 2000 Using T-SQL with Magic Squares Top 20 Viewed Articles How can I uninstall the Microsoft Java Virtual Machine (JVM) from Windows XP? What You Need to Know About Windows Update Services Unneeded Services in Windows XP WinInfo Short Takes: Week of June 7 ... Microsoft Issues Public WMP10 Beta [UPDATED] More Top Viewed Articles Events Central Microsoft TechEd Europe 2004 TechNet Webcast: SQL Server 2005 Beta 2: Using Database Mirroring for High Availability - Level 300 TechNet Webcast: Global Business Process Automation with Siebel and Microsoft SQL Server - Level 200 TechNet Webcast: Database Design and SQL for System Administrators who Script – Level 300 ... T-SQL Black Belt InstantDoc #9020 SQL Server Magazine DOWNLOAD THE CODE: 9020.zip

94. Magic Square PROBLEM magic square. Find a 3 X 3 magic square whre the operation is multiplicationrather than addition and the entries are 9 different numbers. 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 whre the operation is multiplication rather than addition and the entries are 9 different numbers. Return to the EMAT 4600/6600 Page

95. Magic Square Variations Unique matrix positions in magic square pattern rotation and reflection. The consensuscorrectly maintains that there is only one order 3 magic square. http://www.innotts.co.uk/deveritt/magicsquare/variations.htm

a r t D4 transformation images Unique matrix positions in magic square pattern rotation and reflection View the entire image sequence The following explains the problems inherent in deciding whether rotations and reflections should be considered as original magic squares on a fixed matrix, or as a smaller separate number of fixed pattern groups based solely on numerical relation and translated through rotation and reflection. The case is argued for the latter for reasons outlined in the summary below. General notes: This description is written by a digital artist with an interest in magic square and cube patterns, not a mathematician. Illustrations and further explorations of the idea are in preparation. Thanks to Simon Nee at Loughborough University Computer Human Interface Department for his input on refining the formula. Thanks also to Alan Grogono Dave Harper and Charles Kelley (and their magic square websites) who have responded with helpful and informed comments. Summary This inquiry begins with the already obvious number of possible permutations based *only* on rotation and reflection, for any given magic square. The answer is, of course, 8 - the Dn or dihedral group of symmetries for a square where n is the order of regular-sided polygon rotated. However, this raises a further, more fundamental issue of the number of unique positions within a magic square matrix that any number can occupy, once those eliminated by the symmetry group are exculded. Where magic squares are classified as sets of integers in a fixed matrix in which each position is unique and static, the eight rotations and reflections of any magic square pattern within that matrix must be regarded as separate entities. However, the generally held position disreguards these rotations and reflections, as do, perhaps, some attempts to formulate exhaustive formulae for determining magic square permutations. But if each individual square is be treated as a single group where the eight rotations and reflections form a group of symmetries (trivial to the existing method) for that square, there are implications for such formulae. If any number in a magic square is rotated and reflected through the square's group of eight symmetries, it can be seen to occupy either 1, 4 or 8

96. The Magic Squares Puzzle The magic squares Puzzle, see if you can solve it. Java Programmingby Kyle Palmer. The magic squares by Kyle Palmer. The object of http://www.worldkids.net/puzzles/puzzle2.htm

The Magic Squares by Kyle Palmer The object of this puzzle is to get all of the squares to be the same color. Clicking on the square will change the color of the square as well of as change the squares around it. The number of squares and number of colors can be modified, although no-one has ever got past 9 squares with two colors. Hopefully soon there will be a high score save feature as well as a solve feature. We don't know if this puzzle can even be solved with more than two colors. Enjoy! tm Map Site Index Search WKN Unauthorized Reproduction Prohibited

97. Magic Squares Further information. This sort of magic square can be set up for any magic number.For example, let us set up a magic square that has the magic number 42. http://www.questacon.edu.au/html/magic_square.html

Questacon Teachers Outreach Programs Questacon Maths Centre ... Post-visit Activities Maths Centre Activities What is Topology The Handcuffs Puzzle Handcuff Puzzle Images Linking Paperclips ... Bubble Mix Recipe Mathematical idea When adding numbers, it does not matter what order you add in. Materials needed A piece of paper with a 4x4 grid drawn on it and a pen. Demonstrations Start with a 4x4 grid, with the numbers from one to sixteen in it. Choose one number, then cross out the other numbers in the same row and column. (e.g. if you choose 7, cross out 5, 6, 8, 3, 11 and 15) Repeat until you only have four numbers left. If you add these numbers together you will get 34. Further information This sort of magic square can be set up for any magic number. For example, let us set up a magic square that has the magic number 42. Start by choosing eight numbers that add to 42. Let's use 3, 5, 6, 8, 4, 1, 9 and 6. Now draw up a five by five grid. Leave the top right hand corner empty and write four of the numbers in the remaining squares in the top row. Write the remaining four numbers in the first column, leaving the top right corner empty again. Next fill in the grid as an addition table, as shown.

98. Magic Squares The classic form of a magic square is a square containing consecutive numbers startingwith 1, in which the rows and columns and the diagonals all total to the http://home.ecn.ab.ca/~jsavard/math/squint.htm

Home Other Mathematics Magic Squares Magic Squares may be perhaps the only area of recreational mathematics to which many of us have been exposed. The classic form of a magic square is a square containing consecutive numbers starting with 1, in which the rows and columns and the diagonals all total to the same number. I'll have to admit that I was never very much interested by magic squares, as opposed to other mathematical amusements, but a Mathematical Games column in Scientific American by Martin Gardner disclosed some new discoveries in magic squares that are of interest. The only magic square of order 3, except for trivial translations such as reflection and rotation, is: Some magic squares are very simple to construct. Magic squares of any odd order can be constructed following a pattern very similar to that of the 3 by 3 magic square: One can also construct a magic square by making a square array of copies of a magic square, and then adding a displacement to the elements of each copy based on a plan given by another magic square: thus, making nine copies of

99. Pickover, C.A.: The Zen Of Magic Squares, Circles, And Stars: An Exhibition Of S of the book The Zen of magic squares, Circles, and Stars An Exhibitionof Surprising Structures across Dimensions by Pickover, CA, published by...... http://pup.princeton.edu/titles/7131.html

PRINCETON University Press SEARCH: Keywords Author Title More Options Power Search Search Hints E-MAIL NOTICES NEW IN PRINT E-BOOKS ... HOME PAGE The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions Clifford A. Pickover Shopping Cart Reviews Table of Contents Humanity's love affair with mathematics and mysticism reached a critical juncture, legend has it, on the back of a turtle in ancient China. As Clifford Pickover briefly recounts in this enthralling book, the most comprehensive in decades on magic squares, Emperor Yu was supposedly strolling along the Yellow River one day around 2200 B.C. when he spotted the creature: its shell had a series of dots within squares. To Yu's amazement, each row of squares contained fifteen dots, as did the columns and diagonals. When he added any two cells opposite along a line through the center square, like 2 and 8, he always arrived at 10. The turtle, unwitting inspirer of the ''Yu'' square, went on to a life of courtly comfort and fame. Pickover explains why Chinese emperors, Babylonian astrologer-priests, prehistoric cave people in France, and ancient Mayans of the Yucatan were convinced that magic squaresarrays filled with numbers or letters in certain arrangementsheld the secret of the universe. Since the dawn of civilization, he writes, humans have invoked such patterns to ward off evil and bring good fortune. Yet who would have guessed that in the twenty-first century, mathematicians would be studying magic squares so immense and in so many dimensions that the objects defy ordinary human contemplation and visualization?

100. Amof:Info On Magic Squares The result is known as a magic square. Example. For example, all the magic squaresof size 3 by 3 are shown below 2, 7, 6. 9, 5, 1. 4, 3, 8. 2, 9, 4. 7, 5, 3. 6,1, 8. 6, 1, 8. http://www.schoolnet.ca/vp-pv/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.

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