41. Grogono Family Home Page The Magic Carpet approach to understanding Magic Squares by Grog . http://www.grogono.com/Magic/

Home Page About the Site Family Tributes Other Websites ... Contact Resources: Animated Knots Magic Squares Stereo Art Anesthesiology Recruitment ... Health Measurement Welcome Riley Standing on the Beach Revised Again This revision was prompted by having to move the website; the site was rapidly approaching the upper limit for monthly traffic. The traffic is mostly due to the sections on Animated Knots Magic Squares , and Stereo Art . Searches on Google and Yahoo showed that these three sites were at or near the top of their searches! To see where they are on Google at present, click on Google Search:Animated Knots Google Search:Magic Squares , and Google Search:Stereo Art . An additional knot has recently been added, the Cleat Knot , as well as some more examples of Stereo Art. Now Welcome "Grog" (Alan) and Anthea thank you for visiting our Family Website. This website was started in 1996 - which, on the internet seems a long time ago. The website was, originally, going to be mostly about the family. However, as you will have realized from the paragraph above, the website seems to have grown in various different directions. Current News We're still enjoying retirement - although we still seem to be far too busy. Grog still enjoys creating websites. One of his latest is for

42. Allmath.com - Magic Squares Game In the easy level, you are given the numbers beginning from one and must createa magic square. How large would you like the magic square to be on each side? http://www.allmath.com/magicsquare.php

Option Explicit 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 Magic squares have been a fascinating topic in mathematics for centuries. They are formed by filling in all the squares with the numbers starting from one so that the sum of all row, columns, and diagonals is the same. If you know how to play, you can use the square below. Otherwise, read the instructions and fill in the form at the bottom to begin a new game. Numbers to use: Numbers you have used: In this game, you must fill in the magic squares. In the easy level, you are given the numbers beginning from one and must create a magic square. The medium difficulty level is similar, but instead of getting each row, column, and diagonal to add to the same value, each one has a different value which is given to you. In the hard level, you are given random numbers, some positive and some negative, and you must place them in the square to get the sums given to you. To start playing, you first select whether you want a 3 by 3 square, 4 by 4 square, or 5 by 5 square. As the square gets bigger, the puzzle becomes more difficult. Next, you select the difficulty level you wish to play.

43. Randomize Function Sigma(lowbound, Hibound, Func, N, TheSize How large would you like the magic square to be on each side? In the easy level,you are given the numbers beginning from one and must create a magic square. http://www.allmath.com/MagicSquare.asp

< Min then Min = TheArray(j) NumtoSwap = j end if Next if NumToSwap numb then temp = TheArray(NumToSwap) TheArray(numToSwap) = TheArray(numb) TheArray(numb) = temp End if Next End Sub Sub RandomizeArray(size, TheArray) for index = to size - 2 RandomNum = RandBetw(index,size-1) if index size then do Position = int(rnd*size) if TheHintArray(position) Numbers to use: Numbers you have used: ") else response.write(" ") end if next response.write(" ") end sub sub PrintSquare(size, theArray) userDiag1 = sigma(1, sizeOfSquare, "diag1", 1, sizeOfSquare, userArray) userDiag2 = sigma(0, sizeOfSquare-1, "diag2", 1, sizeOfSquare, userArray) for loopVar = 1 to size UserSumCol(loopVar) = sigma(0, sizeOfSquare-1, "column", loopVar, sizeOfSquare, userArray) UserSumRow(loopVar) = sigma((loopVar-1)*sizeOfSquare, LoopVar*sizeOfSquare -1, "row", loopVar, sizeOfSquare, userArray) next response.write(" ") if Hint(position) = 1 then response.write (theArray(position)) response.write(" ") else response.write("

44. Magic Square -- From MathWorld Index of prime magic squaresIndex. magic square What is a magic square? Prime magic square What is aprime magic square? Top, magic square, Tognon Stefano Research. ); //. http://www.astro.virginia.edu/~eww6n/math/MagicSquare.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Algebra Linear Algebra Matrices ... Magic Squares Magic Square A magic square consists of the distinct positive integers such that the sum of the n numbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1952, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacobi 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constant If every number in a magic square is subtracted from another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a "normal" magic square. The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu . A version of the order 4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called . Magic squares of order 3 through 8 are shown above. The magic constant for an n th order general magic square starting with an integer A and with entries in an increasing arithmetic series with difference D between terms is (Hunter and Madachy 1975).

45. Interactive Magic Square 3x3 Hope you like this 3 x 3 magic square!! Find out more about magic squares. ViewBenjamin Franklin s 8x8 magic square in animation. More Teaching Resourcese. http://www-personal.une.edu.au/~lgrunwa2/une/KLAs/maths/magic3x3.html

Hope you like this 3 x 3 Magic square!! Find out more about magic squares. View Benjamin Franklin's 8x8 magic square in animation More Teaching Resourcese

46. Magic Squares And Recursion magic squares. ). First, what is a magic square? A magic square is a matrix ora 2-dimensional grid of numbers. Take the simple case of a 3x3 magic square. http://personal.vsnl.com/erwin/magic.htm

m a g i c Recursion Index Magic Squares In this page, I'm going to show you the permutation-capabilities of Recursion. Permutation means a combination of certain units in all possible orderings. Recursion can be effectively used to find all possible combinations of a given set of elements. This has applications in anagrams, scheduling and, of course, Magic Squares. And if you're interested, Recursion can also be used for cracking passwords. First, what is a magic square? A magic square is a 'matrix' or a 2-dimensional grid of numbers. Take the simple case of a 3x3 magic square. Here's one:- A Magic Square contains a certain bunch of numbers, in this case, 1..9, each of which has to be filled once into the grid. The 'magic' property of a Magic Square is that the sum of the numbers in the rows and columns and diagonals should all be same, in this case, 15. Try making a 3x3 magic square yourself. It's not that easy. If it was easy, try a 4x4 grid with numbers 1..16. And what about 5x5, 6x6...? That's where computers come in! Okay, now, how do we go about programming something

47. Science News For Kids: MatheMUSEments Melancholia, an engraving by the German artist Albrecht Dürer, includes a famousmagic square. The magic square is hanging on the wall to the upper right. http://www.sciencenewsforkids.org/pages/puzzlezone/muse/muse1103.asp

Home Article Archive Agriculture Animals ... Next Site 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. Why pack so much number magic into one square? Astrologers in Dürer's time associated different types of magic squares with the planets, which, in turn, were thought to influence health. The brooding man is suffering from Saturn's "saturnine," or gloomy, influence. He hopes Jupiter's "jovial" four-by-four magic square will draw down, or decrease, Saturn's influence. (Of course this is all absolutely nutters, but that's the history of ideas for you.)

48. Finding Magic Squares Using CCM A method of finding a magic square using CCM is explained here. A method of findinga magic square using CCM. The applet below searches for a magic square. http://www.ff.iij4u.or.jp/~kanada/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

49. NCTM Illuminations An nthorder magic square is a square array of n 2 distinct integers in whichthe sum of the n numbers in each row, column, and diagonal is the same. http://illuminations.nctm.org/lessonplans/6-8/magic/

Pre-K - 2 Across Lessons SWRs Tools i-Maths Inquiry Search Standards Contact Us Home Internet-Based Multi-Day Lesson Plan / 6 - 8 Magic Squares: Discovering Their History and Their Magic Overview: This article explores magic squares from both a historical and mathematical perspective. The mathematical analysis leads into symbolic algebraic representation of the patterns. An activity sheet for students is included. NCTM Publication-Based Lesson Plans are adapted from NCTM's journals. This lesson plan appeared in the April 2001 edition of Mathematics Teaching in the Middle School Journal. Mathematical Learning Objectives: Students will: use operations with integers to solve problems analyze and represent patterns with symbolic rules NCTM Standards: This topic relates to: Number and Operations Algebra Grade Level: Grades 5 - 8 Time: 1-2 periods Web Sites: Web sites used for the Internet Extensions are: Magic Square Game for students Magic Squares/Magic Carpets Home Page Magic Squares Materials : One Student Activity Sheet is included.

50. Chinese Mathematics: Rebecca And Tommy Clearly the Lo Shu is a straight forward magic square in which the integers addedup along any column, row or major diagonal add up to 15 (see diagram 9). The 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.

51. The Anti-Magic Square Project Solving magic squares 5 from each number to give abc 1 4 3 def = 2 0 -2 ghi -3 4 -1 which makes it immediatelyobvious that there is only one possible magic square of order 3, up http://www.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:

52. Magic Square Of Squares magic square of Squares. It s an open question whether there existsa 3x3 magic square comprised entirely of square integers. Before http://www.mathpages.com/home/kmath417.htm

Magic Square of Squares Orthomagic Square of Squares Automedian Triangles and Magic Squares Discordance Impedes Square Magic Return to MathPages Main Menu

53. Just Riddles And More Magic Squares magic squares. Put the numbers in order so that they read 18. The 0 is the empty place. Click on any number next to 0 and they will switch places. http://www.justriddlesandmore.com/magicsquares.html

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: LIKE THIS GAME? SHARE IT WITH A FRIEND.

54. Magic Square, Magic Square. magic square, A nby-n magic square (magic square of order n) contains n rowsand n columns of numbers, which make up its nxn (n squared) elements. http://www.occultopedia.com/m/magic_square.htm

Home Browse Alphabetically Topics Latest Strange News ... Occultopedia An Encyclopedia of the Occult, the Unexplained, Myths and more... Submit an Article Shopping Additions and Updates ... Links Magic Square No Frames Navigation Info A magic square is a square array of numbers with the property that the sum of each row, each column, and each diagonal is the same. Magic squares have been found in ancient writings from many parts of the world, and in some cultures they were thought to possess magical or supernatural powers. A n-by-n magic square (magic square of order n) contains n rows and n columns of numbers, which make up its n x n (n squared) elements. A magic square remains a magic square if the same number is added to each element or if each element is multiplied by the same number. Adding corresponding elements of two magic squares produces another magic square. Related software audio videos books and info Back to M For explicit instructions on how to order the books, videos and other merchandize offered throughout the links on this site, check our ordering information section . Visit our Virtual Shopping Mall , where you can find a large variety of products, including occult related items. We teamed up with Vstore, Amazon.com, and other online merchants to offer our visitors a wide selection of quality products and services. The proceedings from our sales are used to finance the upkeep and maintenance of this site. Support Occultopedia. Make it possible for this valuable source of reference and entertainment to stay online for years to come by shopping through us (you will never pay a penny more than if you had gone directly to the merchant).

55. LoneWolf's Lair: Magic Squares A magic square is an NxN grid that contains all of the numbers from 1 to N 2 (Nsquared)in such a way that the sum of all of the numbers from any row, column http://www.sightspecific.com/~mosh/Tricks/MagicSquare/

The LoneWolf's Lair. . . Presented by Mosh "LoneWolf" Teitelbaum Magic Squares A magic square is an NxN grid that contains all of the numbers from 1 to N (N-squared) in such a way that the sum of all of the numbers from any row, column, or diagonal equals the same number. For example, a 3x3 grid would contain all of the numbers from 1 to 9 (3-squared) such that, were you to add the numbers from any row, column, or diagonal, the sum would equal 15. Go ahead and enter an odd, positive number in the form field below, click the build button, and see for yourself. This application will generate the magic square for the dimension that you enter. That is, if you enter 3, it will display the 3x3 magic square. Dimension: This page was last updated Monday, February 8, 1999. LoneWolf [Section Index] [Home Page]

56. Magic Square Proof SET® Mathematics Mathematical Proof of the magic square by LlewellynFalco. One day, while sitting by myself with a deck of SET® cards http://www.setgame.com/set/proof.htm

Mathematical Proof of the Magic Square by Llewellyn Falco Number[ X Color[ X Symbol[ X Shading[ X So the vector x=[p,q,r,s] completely describes the card. For example: the card with one, red, empty, oval might be Number[ X ] = 1, Color[ X ] =1, Symbol[ X ] = 1, Shading[ X ] = 1, or x = [1,1,1,1,]. For shorthand, I use the notation C x to represent the card. Where C x = Number[ X ] , Color[ X ] , Symbol[ X ], Shading[ X ], and x = [p,q,r,s]. If I wanted to make the third card which makes a set from two cards Ca and Cb, I would have the card C (ab) where ab = [a b , a b , a b , a b and the rule for the operator is: If an = bn, then bn = xn and an = xn If an bn, then bn xn and an xn For Example: 1*1=1, 1*2=3, 1*3=2 Here are some basic theorems in this group, linked to their proofs: a n b n b n a n a n b n c n a n b n c n a n c n a n b n a n c n b n a a b b The Square So let us begin by choosing any three cards: a, b, and c, and placing them in positions 7, 5, 9. C c C a C b Now we need to fill in the blanks for the remaining cards. Starting with card 8; it needs to complete the set with the cards C a and C b . We now look at the multiplier. The new card will be the product of C a operating on C b which is C ab . Likewise, filling in slots 1 and 3 leaves us with the square below.

57. Set - How To Make A Magic Square Of Set What you see here is a magic square, much like the addition and subtraction squaresyou may have used as a child. Any line on the magic square yields a set. http://www.setgame.com/set/magicsquare.htm

Set Mathematics Magic Squares What you see here is a magic square, much like the addition and subtraction squares you may have used as a child. color shape number of objects, and shading . The rules state for each property, they must all be equal, or all different. For example, if we look at the top row of the square, we see three different colors, three different shapes, three different numbers, and three different types of shading within the objects. Need more examples? Any line on the magic square yields a set. Constructing a magic square may seem complex at first glance, but in reality anyone can make one by following this simple process: Choose any three cards that are not a set. (It will work with a set but the square becomes redundant) For example, we will choose these: Now place these three cards in the #1, #3, and #5 positions in the magic square. Using our powers of deduction, we can conclude that in order to create a set in the first row, the #2 card needs to have a different color, different shape, same number, and same shading as the #1 and #3 cards. That leaves us with a solid purple oval. The rest of the square can be completed in the same way, giving us the following magic square: A few examples will convince you that this method works. Not only does the magic square work but it can be theoretically proven through a

58. The JavaScript Source: Games: Magic Squares Home Games magic squares Here is a fun JavaScript game. Try to put the squaresback in order. Get the squares back in order and you win! magic squares. http://javascript.internet.com/games/magic-squares.html

Home Games Magic Squares Here is a fun JavaScript game. Try to put the squares back in order. The '0' repesents 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: The JavaScript Source: Games: Magic Squares Simply click inside the window below, use your cursor to highlight the script, and copy (type Control-c or Apple-c) the script into a new file in your text editor (such as NotePad or SimpleText) and save (Control-s or Apple-s). The script is yours! Did you use this script? Do you like this site? Please link to us! document.write(''); Sign up for the JavaScript Weekly We'll send this script to you! (just click "Send it!" once!) Contact Us! JavaScript Forum Submit Your Script! JupiterWeb networks: Search JupiterWeb: Jupitermedia Corporation has four divisions: JupiterWeb JupiterResearch JupiterEvents and JupiterImages Legal Notices Licensing Reprints ... E-mail Offers

59. The Magic Square next up previous Next E 6 Up Exceptional Lie Algebras Previous F 44.3 The magic square. This goes by the name of the `magic square . 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 [

60. Magic Square Students opened a file with the following spreadsheet. They learnedhow to enter a formula in a spreadsheet so that the totals would http://www.joannegoodwin.com/technology/5th/magic/

Students opened a file with the following spreadsheet. They learned how to enter a formula in a spreadsheet so that the totals would be calculated as they entered the digits. To enter the formula for finding the sum: Click in the cell where you want sum. For example: click to highlight cell A4. In the text entry bar, type an equal sign. All formulas begin with this symbol. Then click in the cells you want to total. For our example: click in cells A1, A2, and A3. Notice the formula in the text entry bar. When you press return you will enter the formula and the computer will make your calculations. Until digits are entered in the cells your total will be zero. Follow this procedure for each row and column. Back to the CRS Tech Lab

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