61. Magic Square 3, 20, 7, 18, 17. 9, 15, 12, 4, 25. 21, 23, 14, 5, 2. 8, 6, 19, 22, 10. 24,1, 13, 16, 11. magic square's Page. Japanese page 4x4 magic square. 6x6 MagicSquare. Issei Numata http://www.guru.gr.jp/~issei/msq/

Magic Square's Page Japanese page 4x4 Magic Square 6x6 Magic Square Issei Numata

62. 4x4 Magic Square In this page, I show mysterious aspect for 4x4 magic square. If one magic square'sdiagonals line satisfy this condition, the magic square is panmagic square. http://www.guru.gr.jp/~issei/msq/msq4.html

4x4 Magic Squares There are 880 magic squares. (including reflection, rotation, there are 7040 magic squares). If you'd like to show detail, please show Magic Squares page. In this page, I show mysterious aspect for 4x4 magic square. Bad Couple '7' never neighbor on '8'. '9' never neighbor on '10'. However, they can neighbor diagonally. Sum of Corner, Sum of Center Sum of corner is 34 in any magic squares. Sum of center is 34 in any magic squares. This mean sum of edge is 68 and sum of facing edge is 34. Fig. 1 Because, 2 sum of diagonal lines are 68, so sum of colored cells (Fig. 2) is 68. If sum of green cells and sum of purple cells are equal. sum of green cells is 34 and sum of purple cells is 34. Fig.3. shows sum of central cells and sum of corner cells are equal. Because U-V-W = 0, sum of central cells and sum of corner cells are equal and this results sum of central is 34 and sum of corner cells is 34. Fig. 2 U V W X Y Fig. 3 Sum of Any 2x2 Square Sum of 2x2 square which does not neighbor is equal (But, its total may not be 34.) Fig.5 shows why sum of 2x2 square which does not neighbor is equal.

63. Magic Square magic square, Shown on the left is a 258 x 258 magic square, displayedwith the Matlab default colormap (jet). Before normalization http://www.cs.dartmouth.edu/~farid/research/magic.html

Magic Square Shown on the left is a 258 x 258 magic square, displayed with the Matlab default colormap (jet). Before normalization, every row and column of this image sum to 8586885. Home Research

64. MAGIC SQUARE Inc. The summary for this Japanese page contains characters that cannot be correctly displayed in this language/character set. http://www.magicsquare.co.jp/

65. 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:

66. Magic Square 2 Evolving a magic square using Genetic Algorithms. simple integer representationof parameter = corresponding number in magic square;. quality function. http://www.dcs.napier.ac.uk/~benp/summerschool/jdemos/herdy/magic_problem2.html

Evolving a Magic Square using Genetic Algorithms The goal is to arrange the numbers from 1 to N^2 within a NxN grid in such a way that the sum of all rows, the sum of all columns and the sums of both diagonals become equal, i.e. the goal is to find a true magic square. GA - interface help Genetic Algorithms Theory should be an applet here C hromosome (genotype representation) The magic square is encoded using an array of n*n bitvectors each with a size of ceil(ld(n*n)). // 4x4 magic square numbers 1..16 10000..00001 i.e. bit-order 01234 [ [11010][10110][00001][10100] // 11 13 16 5 [01100][11100][10000][11000] // 6 7 1 3 [01000][00010][10010][00110] // 2 8 9 12 [01110][00100][11110][01010] ] // 14 4 15 10 A problem-specific crossover -operator is used which could be refered to as permutation operator. The operator is unary. That is each of the two crossover partners is handled individually. If m is the number of crossover-points than the specific crossover operator simply 'swaps' m pairs of values thus producing another permutation of 1..n^2 (n × n magic square). A repair -operator is used. If used, the mutation operator will lead to 'illegal' magic squares. That is, the magic squares contain a single number more that once or don't contain it at all. Additionally it may produce 'out of range'-values i.e. values smaller than 1 or bigger than n^2. Thus after crossover and mutation a repair operator is used to produce 'well shaped' magic squares again.

67. Magic Square's Family version francaise. magic square's Family. These are otherkinds of squares whose definition may be very different http://perso.club-internet.fr/spip/carcousinseng.html

M ag i c sq u a re 's F a mi ly These are other kinds of squares whose definition may be very different : Latin squares Eulerian squares Most-perfect magic squares Magic cubes Latin square This is another square filled with integers from 1 to n. Each of them can be seen n times and has to be only one time on each line (vertical, horizontal or diagonal). These are 2 examples of this sort of squares considering n=3. Back to the index Eulerian square If you try to superpose the two latin squares above keeping exactly the same order, so as to build the squares of the pairs, you can notice that each pair is represented only one time. These squares of integers'pairs are called eulerian squares (referencing to the Swiss mathematician Leonhard Euler) but they are sometimes also called Greek squares. Back to the index Most-perfect magic squares In these squares with an even order, the magical sum of lines also exists in breaked diagonals. Besides that, the sums of the little squares included in the big one are always the same. This applet under (only for Java 1.1 compatible browsers, in French for I am French as you understood reading my awfull English...) shows a good example of a most-perfect magic square with an order of 8 which you can see all that magic sums on.

68. Semi-Magic Square Knight Tours Semimagic square Knight Tours. Also, 81 is 9 squared in which 9 represents thefirst known magic square (3x3 square or 9 squares) from about 2200B.C. http://www.borderschess.org/KTsemi.htm

Semi-Magic Square Knight Tours Without the aid of a computer, I've easily created many more Knight Tours and even a Semi-Magic Square Knight's Tour where all the rows and columns add up to the same number. In fact, every other vertical pair of numbers adds up to 49 and 81 respectively while the four major quadrants each add up to 520 and the four main sets of 2x2 squares in each quadrant add up to 130. The numbers 49 and 81 are significant since 49 is 7 squared in which 7 represents the shape of the Knight's move. Also, 81 is 9 squared in which 9 represents the first known Magic Square (3x3 square or 9 squares) from about 2200B.C. With a little bit of research, you will find that others such as William Beverley have created similar Semi-Magic Square Knight Tours. Unlike William Beverley, I've taken an additional step by creating a simple game that will allow anyone to easily make beautiful mathematical Knight's Tour art. Please let me know if you are interested in my other Knight Tours, Knight Tour game, or my basic mathematical findings that provides such easy creation of Knight Tours. Here are a couple semi-magic Knight Tours that are almost identical. Notice how the first and third columns from the left are interchangeable. I thought I was quite clever coming up with these patterns until I found out that William Beverley designed the first such tour with the same path pattern of the knight around 1847-1848. Check out

69. Magic Square - 4 magic square 4 / Magisk kvadrat - 4. Oh! Did your boss come back - run. ReturnÅh! Kom din chef tilbage - løb. Retur. magic square - 4 / Magisk kvadrat - 4. http://www.danbbs.dk/~erikoest/magicsq4.htm

Magic Square - 3 / Magisk kvadrat - 3 Magic Square - 5 / Magisk kvadrat - 5 Bottom of This Page Sorry! / Beklager! You must have a script-capable browser to view this document correctly. / Du skal benytte en script-egnet browser for at se dette dokument korrekt. Magic Square - 4 / Magisk kvadrat - 4 Fill in the squares with one of each of the numbers from 1 to 16 so that the sum of the numbers in any row, column, or diagonal is the same. / Message / Besked: Oh! Did your boss come back - run Magic Square - 4 / Magisk kvadrat - 4 Click on the button "Calculate / Beregn" when your are ready - and wait while computer calculates the result. / Klik på knappen "Calculate / Beregn" når du er færdig - og vent medens computeren beregner resultatet. "Message / Besked" shows messages from the computer. / "Message / Besked" viser beskeder fra computeren. "Restart / Genstart" resets to play again. / "Restart / Genstart" nulstiller for fornyet spil. "Solution / Løsning" shows the sum of the Magic Square, and if you want you can see an example of a correct completed Magic Square. Other solutions can be correct too. / "Solution / Løsning" viser det magiske kvadrats sum, og hvis du ønsker det, kan du se et eksempel på et rigtigt udfyldt magisk kvadrat. Der findes også andre løsninger, som er rigtige.

70. Magic Square - 5 magic square 5 / Magisk kvadrat - 5. Oh! Did your boss come back - run. ReturnÅh! Kom din chef tilbage - løb. Retur. magic square - 5 / Magisk kvadrat - 5. http://www.danbbs.dk/~erikoest/magicsq5.htm

Magic Square - 3 / Magisk kvadrat - 3 Magic Square - 4 / Magisk kvadrat - 4 Bottom of This Page Sorry! / Beklager! You must have a script-capable browser to view this document correctly. / Du skal benytte en script-egnet browser for at se dette dokument korrekt. Magic Square - 5 / Magisk kvadrat - 5 Fill in the squares with one of each of the numbers from 1 to 25 so that the sum of the numbers in any row, column, or diagonal is the same. / Message / Besked: Oh! Did your boss come back - run Magic Square - 5 / Magisk kvadrat - 5 Click on the button "Calculate / Beregn" when your are ready - and wait while computer calculates the result. / Klik på knappen "Calculate / Beregn" når du er færdig - og vent medens computeren beregner resultatet. "Message / Besked" shows messages from the computer. / "Message / Besked" viser beskeder fra computeren. "Restart / Genstart" resets to play again. / "Restart / Genstart" nulstiller for fornyet spil. "Solution / Løsning" shows the sum of the Magic Square, and if you want you can see an example of a correct completed Magic Square. Other solutions can be correct too. / "Solution / Løsning" viser det magiske kvadrats sum, og hvis du ønsker det, kan du se et eksempel på et rigtigt udfyldt magisk kvadrat. Der findes også andre løsninger, som er rigtige.

71. 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, CDNow 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).

72. Magic Square. The American Heritage® Dictionary Of The English Language: Fourth magic square. The American Heritage® Dictionary of the English LanguageFourth Edition. 2000. Edition. 2000. magic square. NOUN http://www.bartleby.com/61/62/M0026250.html

Select Search All Bartleby.com All Reference Columbia Encyclopedia World History Encyclopedia World Factbook Columbia Gazetteer American Heritage Coll. Dictionary Roget's Thesauri Roget's II: Thesaurus Roget's Int'l Thesaurus Quotations Bartlett's Quotations Columbia Quotations Simpson's Quotations English Usage Modern Usage American English Fowler's King's English Strunk's Style Mencken's Language Cambridge History The King James Bible Oxford Shakespeare Gray's Anatomy Farmer's Cookbook Post's Etiquette Bulfinch's Mythology Frazer's Golden Bough All Verse Anthologies Dickinson, E. Eliot, T.S. Frost, R. Hopkins, G.M. Keats, J. Lawrence, D.H. Masters, E.L. Sandburg, C. Sassoon, S. Whitman, W. Wordsworth, W. Yeats, W.B. All Nonfiction Harvard Classics American Essays Einstein's Relativity Grant, U.S. Roosevelt, T. Wells's History Presidential Inaugurals All Fiction Shelf of Fiction Ghost Stories Short Stories Shaw, G.B. Stein, G. Stevenson, R.L. Wells, H.G. Reference American Heritage Dictionary magic number ... BIBLIOGRAPHIC RECORD The American Heritage Dictionary of the English Language: Fourth Edition.

73. Magic Square. The American Heritage Dictionary Of The English Language: Fourth E magic square. The American Heritage Dictionary of the English LanguageFourth Edition. 2000. 2000. magic square. top the SATOR letter square. http://www.bartleby.com/61/imagepages/A4magics.html

Select Search All Bartleby.com All Reference Columbia Encyclopedia World History Encyclopedia World Factbook Columbia Gazetteer American Heritage Coll. Dictionary Roget's Thesauri Roget's II: Thesaurus Roget's Int'l Thesaurus Quotations Bartlett's Quotations Columbia Quotations Simpson's Quotations English Usage Modern Usage American English Fowler's King's English Strunk's Style Mencken's Language Cambridge History The King James Bible Oxford Shakespeare Gray's Anatomy Farmer's Cookbook Post's Etiquette Bulfinch's Mythology Frazer's Golden Bough All Verse Anthologies Dickinson, E. Eliot, T.S. Frost, R. Hopkins, G.M. Keats, J. Lawrence, D.H. Masters, E.L. Sandburg, C. Sassoon, S. Whitman, W. Wordsworth, W. Yeats, W.B. All Nonfiction Harvard Classics American Essays Einstein's Relativity Grant, U.S. Roosevelt, T. Wells's History Presidential Inaugurals All Fiction Shelf of Fiction Ghost Stories Short Stories Shaw, G.B. Stein, G. Stevenson, R.L. Wells, H.G. Reference American Heritage Dictionary Illustrations ... BIBLIOGRAPHIC RECORD The American Heritage Dictionary of the English Language: Fourth Edition. magic square top: the SATOR letter square bottom: Melencolia I (Source: Precision Graphics) The American Heritage CONTENTS INDEX ILLUSTRATIONS BIBLIOGRAPHIC RECORD ... major scale Search Amazon: Click here to shop the Bartleby Bookstore Welcome Press Advertising ... Bartleby.com

74. Magic Square magic square -. the X edition time is come. http://www.geocities.co.jp/AnimeComic/5164/

- Magic Square - the X edition time is come. contents rireki profile check zakki ... round.5 "zakki"...2003/03/07(Fri) ..sorry,but,this site is japanese only! "Magic Square-2nd"‚ւ悤‚±‚»B ‚±‚±‚Í"‹C‚Ü‚®‚ê‚È"ƒTƒCƒg‚Å‚·B ƒL[ƒ[ƒh‚́A"“à—Ö"A"“dŒ‚"A"ƒKƒ“ƒKƒ“"‚Å‚·B ‚‚܂é‚Æ‚±‚ëA"‹C‚Ü‚®‚ê"‚È‚Ì‚Å‚·B – •PI—¹ŒãA.hack‚É–ÚŠo‚ß‚½Žál‚½‚¿‚Ì °‚Ì‹©‚Ñ‚ð‚ЂƂ±‚Ƃɍž‚ß‚éƒTƒCƒgB iƒ’·ƒFj since.2001/12/09 ŠÇ—lFms2

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

76. Magic Squares At Math Cats: Fun, Easy, And Interactive! See how many ways the Durer magic square adds up to the same number. Thisspecial magic square adds up to the same number in all of these ways. http://www.mathcats.com/explore/puzzles/magicsquare.html

contents math cats explore Magic Squares In magic squares, each row and each column and each diagonal adds up to the same number. But look at the green markings at the top of this page! This special magic square adds up to the same number in all of these ways. (Don't take our word for it! Use your mental math and see for yourself!) It is called a "pan-magic square." We didn't invent this magic square. A famous German artist, Albrecht Dürer, included it in an engraving called "Melancholia" in the year 1514! The most magic thing of all is that you can see the year "1514" in the middle of the bottom row. Pass your mouse over the square to see what it looked like in the 1514 engraving! Click on the square to visit our Math Cats' magic square activities.

77. Magic Square QuizCards. Mind Stretching, Trivia Quizdom, Jigsaw Palace, Quiz Cards.magic square 5 x 5. Complete this square with integers from 6 to 25. http://www.queendom.com/mindgames/mindstretching/puzzles/g20.html

78. Magic Square magic square Instructions The object of the game is to create a magic square ,that is, by clicking the game board cells to form one large red square. http://www.prosanityworld.com/games/square.html

Magic Square: Instructions The object of the game is to create a "Magic Square", that is, by clicking the game board cells to form one large red square. At the start of the game, each cell will be randomly colored blue or red. Clicking on a cell causes it to change color, as well as several of its neighbors. You must discover the pattern of clicks to form the "Magic Square". The game ends when the center cell is colored blue, and all of the surrounding cells have been colored red. Back

79. EVA Magic Square The summary for this Japanese page contains characters that cannot be correctly displayed in this language/character set. http://chearts.g--z.com/evaland/

80. Melancholia Magic Square Melancholia magic square. The most famous 4 x 4 magic square is the socalledMelancholia magic square. The magic constant of the square is 34. http://www.camosun.bc.ca/~jbritton/jbmelancholia.htm

Melancholia Magic Square Melancholie which depicts the indecision of the intellectual. The two numbers, 15 and 14, in the centre of the bottom row date the engraving. Durer never explained the rich symbolism of this masterpiece, but most authorities agree that it depicts the sullen mood of the thinker unable to engage in action. (In the Renaissance, the melancholy temperament was thought characteristic of creative genius.) Unused tools of science and carpentry lie in disorder around the disheveled, brooding figure of Melancholy. There is nothing in the balance scales, no one mounts the ladder, the sleeping hound is half-starved, the winged cherub waits for dictation while time is running out in the hourglass above. The wooden sphere and curiously truncated stone tetrahedron suggest the mathematical base of the building arts. The lunar rainbow arching over what appears to be a comet may signify the hope that the somber mood will pass. The magic constant of the square is 34. In other words, the sum of the numbers in each of the four rows, the four columns, and the two main diagrams is 34. However, this square has an amazing number of other "magic" properties - including many more ways of attaining the constant sum of 34. Can you discover some on your own? Click here for the solution.

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