41. Freudenthal-Tits Magic Square Tony Smith's Home Page. FreudenthalTits magic square Joseph M. Landsberg approachesthe Freudenthal-Tits magic square from an Algebraic Geometry Point of View. http://www.innerx.net/personal/tsmith/FTsquare.html

Tony Smith's Home Page Freudenthal-Tits Magic Square: Here are some approaches from other points of view: Geoffrey Dixon John Baez - Division Algebras C.H. Barton and A. Sudbery J. M. Landsberg - Algebraic Geometry ... References The E6-E7-E8 structures are based on the Freudenthal-Tits Magic Square , which shows relationships between division algebras and matrix algebras. In particular: D ivision algebras define the rows of the Magic Square; J ordan algebras define the columns of the Magic Square; and Lie algebras Lie algebras L formed by the rule: L = Der(A) + (A0xJ0) + Der(J) where Der means derivation, + is direct sum, x is tensor product, A0 are the pure imaginary elements of A, R0=S0, C0=S1, Q0=S3, O0=S7, (here Sn means the algebra of tangent vectors to an n-dim sphere, and S0,S1,S3 are Lie algebras Geoffrey Dixon shows how to construct F4, E6, E7, and E8 from a slightly different point of view F4 = Spin(8) + SO(3) + 3x7 = 28 + 3 + 3x7 = 52 E6 = Spin(8) + SU(3) + 6x7 = 28 + 8 + 6x7 = 78 E7 = Spin(8) + Sp(3) + 12x7 = 28 + 21 + 12x7 = 133 E8 = Spin(8) + F4 + 24x7 = 28 + 52 + 24x7 = 248 To get from Geoffrey Dixon's construction to my construction

42. 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.

43. 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

44. Magic Square magic square. A version of the order 4 magic square with the numbers 15 and 14 inadjacent middle columns in the bottom row is called Dürer's magic square. http://mathworld.pdox.net/math/m/m029.htm

Magic Square A (normal) magic square consists of the distinct Positive Integers such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same Magic Constant 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 th order magic square starting with an Integer and with entries in an increasing Arithmetic Series with difference between terms is (Hunter and Madachy 1975). If every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. Squares which are magic under multiplication instead of addition can be constructed and are known as Multiplication Magic Squares . In addition, squares which are magic under both addition and multiplication can be constructed and are known as Addition-Multiplication Magic Squares (Hunter and Madachy 1975).

45. Magic Square By Coats And Clark magic square POTHOLDER. Remove contrasting color thread. HOW TO PULLPOCKET INTO A magic square Place pocket on a flat surface. http://home.fuse.net/SouthwestOhioCrochetGuild/MagicSq.htm

MAGIC SQUARE - POTHOLDER DUPLICATED BY PERMISSION MATERIALS GAUGE - BE SURE TO CHECK YOUR GAUGE. USE ANY SIZE HOOK THAT WILL OBTAIN THE GAUGE GIVEN. 4 single crochet = 1 inch; 7 rounds = 2 inches POCKET Chain 40, having 4 chain stitches to every inch. ROUND 1: Make 3 single crochet in 2nd chain from hook, place a safety pin in 2nd single crochet of this group for marker; single crochet in next chain and in each chain to last chain, make 3 single crochet in last chain, place a safety pin in 2nd single crochet of this group; working along opposite side of starting chain single crochet in next chain and in each of remaining 36 chain stitches; do not join round. Place a contrasting color thread between last single crochet and next single crochet and carry thread up on each round to indicate beginning of each round. There are 39 single crochet between marked stitches. ROUND 2: Single crochet in BACK loop of next single crochet and in BACK loop of each single crochet around. Repeat Round 2 until piece is half the length of starting chain. At end of last round worked, join with slip stitch to next single crochet. Leaving a 3 inch end, cut yarn and draw end through loop on hook, then draw end through back of a few stitches to secure. Remove contrasting color thread. HOW TO PULL POCKET INTO A "MAGIC SQUARE"

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

Home Introduction About Delphi Feedback ... Site Search (bottom of page) Available Now (Click a level below to expand program list if it is collapsed - MS Internet Explorer only) 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 Akerue Brute Force Countdown Plus! ... WordStuff 2 Contact Feedback Send an e-mail with your comments about this program (or anything else). Search WWW Search delphiforfun.org Problem Description Durer's Magic Square is contained in a famous copper engraving, "Melancholia", created in 1514 by German artist Albrecht Durer. 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 Durer 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 Durer in these studies. In 1514 he created the copperplate engraving names "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.)

47. Magic Square The magic square Intro Page is an excellent synopsis of this field. Here is the mostfamous magic square, from Albrecht Durer's Melancholia. He did it in 1514. http://www.mathpuzzle.com/masquare.htm

Welcome to mathpuzzle.com Magic Squares Several palindromic magic squares can be found at The World of Numbers . The World of Palindromic numbers would be more appropriate. The Magic Square Intro Page is an excellent synopsis of this field. Here is the most famous magic square, from Albrecht Durer's Melancholia. He did it in 1514.

48. Steenz Software magic square Creator v1.4.16 magic square Creator Screenshot Steenz' magic squareCreator lets you create magic squares with just a few mouseclicks. http://home.wanadoo.nl/steenzsoftware/magicsquarecreator/

:: home :: last updated: 02-06-2002 Information About us Contact us News FAQ ... Thanks to... Products ASX Maker Magic Square Creator Downloads ASX Maker Magic Square Creator Magic Square Creator v1.4.16 Steenz' Magic Square Creator lets you create magic squares with just a few mouse-clicks. It's very easy to use, even for beginners. It includes (as a bonus) a full description of the algorithms that are used, so you can learn to understand a magic square. It also includes a help file, with all the information you need to install the program, to run the program, to create magic squares and to uninstall the program. General Program Description A Magic Square is a square filled with different numbers in a way that the sums of ALL columns, rows and diagonals are the same. A simple and small example is: In this example of 5x5, the sum of ALL diagonals, rows and columns is 65. For more information see section 2.3 about the algorithms used. Features create magic squares in just a few clicks the program comes with an easy to use help file the help file contains all the information you need to understand the algorithms used choose the size of the square and the starting value create squares of unlimited sizes, depending on your computers memory

49. Steenz Software magic square Creator download page Here you can download various versionsof magic square Creator. It is recommended to download http://home.wanadoo.nl/steenzsoftware/magicsquarecreator/download.html

:: home :: last updated: 02-06-2002 Information About us Contact us News FAQ ... Thanks to... Products ASX Maker Magic Square Creator Downloads ASX Maker Magic Square Creator Magic Square Creator download page Here you can download various versions of Magic Square Creator. It is recommended to download version 1.4.16 with the VB Runtimes, but there are also versions without the VB Runtimes, old versions and a newer beta version available (read more about them here Latest recommended release: Download Magic Square Creator v1.4.16 with VB Runtimes from ZDNet - 1,8 MB (Please download from ZDNet so I can keep track of how often this file is downloaded.) Download Magic Square Creator v1.4.16 with VB Runtimes directly from this site - 1,8 MB Download Magic Square Creator v1.4.16 without VB Runtimes - 52 kB Public beta-versions (possibly unstable): There are no beta version released yet. Older versions: Download Magic Square Creator v1.4.15 with VB Runtimes - 1,46 MB Download Magic Square Creator v1.4.15 without VB Runtimes - 28 kB Explanation This is the download page for Magic Square Creator.

50. 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

51. Magic Square Français / Anglais; Environnement requis Terminal en mode texte ou X11; http://logiciels-libres-cndp.ac-versailles.fr/pedagogie/magic-square.html

Magic Square Informations générales : Auteur : Paul Landes Licence : GPL Langue : Français / Anglais Environnement requis : Terminal en mode texte ou X11 Site web : ftp://ftp.ac-grenoble.fr/ge/mathematics/magicsqr-1.5.tgz Version testée : Usages pédagogiques : Magicsqr est un petit programme en ligne de commande qui produit des carrés magiques de taille quelconque. La sortie se fait sous forme d'une table affichée au format ASCII (caractères purs), ou sous forme d'un tableau html ou d'un fichier TeX. Xmagicsqr est une interface graphique optionnelle pouvant être associée à magicsqr pour faciliter son usage et la visualisation des résultats. Les carrés magiques sont un thème possible d'étude en primaire, au collège ou au Lycée: arithmétique, résolution d'équations... Magicsqr peut (modestement) aider à la préparation d'un document comportant des exemples de carrés magiques. Retour Sommaire général

52. Mike's Magic Place - MAGIC SQUARE Translate this page My magic square. If you want to create the 'PERFECT magic square',enter a number from 23 - 100 and press the Calculate button Wenn http://members.aol.com/mdormann/tricks/vernon.html

My Magic Square If you want to create the 'PERFECT MAGIC SQUARE', enter a number from 23 - 100 and press the Calculate button: Wenn Sie das 'PERFEKTE MAGISCHE QUADRAT' erzeugen wollen, dann geben Sie eine Zahl zwischen 23 und 100 ein und klicken auf den Button Berechnung Now let's check which fields will add up to the desired number. The rows: The columns: The 4 squares in the corners: The center square: The diagonals: The 4 corners: ... and now let's check for symmetry:

53. A Magic Square Activity A magic square Activity. Let Your Students Learn and Enjoy Math Through the MultipleIntelligences! A magic square MATH ACTIVITY starts at the white page. http://www.markwahl.com/Magic-Square-Activity.htm

A Magic Square Activity Let Your Students Learn and Enjoy Math Through the Multiple Intelligences! If you are searching for any of the following topics: Magic Square Classroom Math Activities Multiple Intelligences Math Homeschooling Math Just click the above and you'll get to a power-packed reference that will take you to a new world of math teaching. SCROLL DOWN to the starting line TO FIND THE FIRST PAGE OF THE FREE multi-intelligence LESSON on magic squares from the book Math for Humans: Teaching Math Through 8 Intelligences [PLEASE BE PATIENT with downloading time sharpness of the page requires it] (You'll find even more possibilities for math-in-color by clicking Mark Wahl Math Learning/Teaching Services! BACKGROUND: The magic squares have been known since antiquity to have the uncanny ability to organize the first 9 or 16 (and beyond) counting numbers in such a beautiful coordination that many astounding math harmonies result. Youths are charmed by them to some extent from a numerical point of view but the addition of more approaches for more intelligences like the VISUAL/SPATIAL, the LINGUISTIC and the INTERPERSONAL, and even the MUSICAL, make them irresistably fascinating and mysterious. A Magic Square MATH ACTIVITY starts at the white page.

54. A Pagan Gathering: WulfeWeb.com AddMe Website Promotion Tool. Generate a Planetary magic square. Generatea magic square for Saturn. http://www.wulfeweb.com/modules.php?name=MagicSquare

55. 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

56. Magic Square - Brain Boosters - DiscoverySchool.com Science Fair Central. Browse the Brain Boosters Gallery. Number Math Play MagicSquare, Then, for a real challenge, make up your own magic square. 12, 16. 4, 8. http://school.discovery.com/brainboosters/numberplay/magicsquare.html

All DiscoverySchool Students Teachers Parents Magic Square Fill in the missing numbers so that the numbers in every row, down, across, and diagonally, will add up to 30. Then, for a real challenge, make up your own magic square. Site Map About Us Feedback Discovery.com

57. Allan Adler - What Is A Magic Square? A Math Forum Web Unit Allan Adler's. What is a magic square? The simplestmagic square is the 1x1 magic square whose only entry is the number 1. http://www.swiss.ai.mit.edu/~adler/MAGIC/adler.whatsquare.html

A Math Forum Web Unit Allan Adler's What is a Magic Square? Suzanne's Magic Squares Multiplying Magic Squares: Contents Exploring the Math Definition, Special Properties A magic square is an arrangement of the numbers from to n^2 (n-squared) in an n x n matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same. It is not hard to show that this sum must be n(n^2+1)/2 The simplest magic square is the magic square whose only entry is the number The next simplest is the magic square and those derived from it by symmetries of the square. This square is definitely magic and satisfies the definition given above. The (or, what is essentially the same, the 4x4 magic square I use) has many interesting special properties that are not shared by magic squares in general. They are so interesting that they are often pointed out when this square is presented. That is good, but can sometimes lead to misunderstandings as to which is the meat and which is the gravy. The meat is the definition I gave above. The gravy (or some of it), suggested by Jerome S. Meyer in his book, Fun with Mathematics

58. What Is A Magic Square? What Is A magic square? The first question, for those who have not comeacross magic squares before, must be, What is a magic square? http://freespace.virgin.net/mark.farrar1/msqwht01.htm

What Is A Magic Square? The first question, for those who have not come across Magic Squares before, must be, "What is a Magic Square?" There are two types of Magic Square: Alphabetic Numeric Alphabetic Magic Squares There are two types of Alphabetic Magic Square: Word Squares Latin Squares Word Squares The early ones included a series of letters, arranged in a square, to spell certain words. One of the most well-known of this variety, throughout the Western world at any rate, is the one shown to the right: Note that the five Latin words appear in the same order both horizontally and vertically, and that the five rows may also be read palindromically. They also form a sentence - "Arepo, the sower, guides the wheels at work". Many people also think that it has a Christian significance, because it contains, jumbled up, the words "pater noster", which, of course, means "our father", together with the letters "a" and "o", which are the Greek letters Alpha and Omega, used of God, for example, in Revelations. These sorts of Word Square frequently occur in puzzle magazines today.

59. Magic Squares- Links The following links to other magic square pages may be of interest A Very Verymagic square by Harm Derksen; AITLC Guide to magic squares; allmath.com; http://freespace.virgin.net/mark.farrar1/msqlnk01.htm

Magic Squares - Links The following links to other Magic Square pages may be of interest: A Very Very Magic Square by Harm Derksen AITLC Guide to Magic Squares allmath.com AMOF - Information on Magic Squares Anti-Magic Square Project ... Bogdan Golunski 's site Bookmarks by Kenth Engø Chinese Mathematics Feng Shui Magic Square Freudenthal-Tits Magic Square ... Magic Square (and Five Pennies) Magic Square article in Encarta Magic Square by Keith Hung Magic Square by Questacon Maths Centre Magic Square - from Eric Weisstein's World of Mathematics Magic Square Generator Program (in French) Magic Square Java Game Magic Square Links (Paul Revere Middle School) Magic Square Links (Graveney School) Magic Square Patterns ... Magic Squares by Suzanne Alejandre Magic Squares by Holger Danielsson Magic Squares by Kristin DeFalco Magic Squares by Sherry Douglas Magic Squares by Orland Hooge Magic Squares by Dr. Kanada Magic Squares Magic Squares by Mutsumi Suzuki Magic Squares And Cubes by Aale de Winkel Magic Stars, Squares and Other by Harvey Heinz Magische Quadrate (in German) by Peter Wilker Mark Swaney On The History Of Magic Squares Mathematical Resources (including Magic Square links) by Bruno Kevius MathPages: Number Theory MATH190 Magic Squares Lecture Millennium Penny Magic Square ... More Magic Squares by Paul Pasles Most Perfect Pandiagonal Magic Squares Multiple Magic Squares Pivari Home Page Psychic Stunts by Michael Daniels Rotas Opera Tenet Arepo Sator Magic Square Sacred Geometry: The Magic Square of the Sun Secrets Of Pi ... Solve The Classic Magic Square from Omega

60. Magic Square 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

Outreach Contents: NRMA RoadZone Photonics Questacon Balloon Questacon Maths Centre ... Teacher Workshops Maths Centre Activities What is Topology The Handcuffs Puzzle Handcuff Puzzle Images Moebius Strip ... 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.

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