61. Ivars Peterson's MathLand: More Than Magic Squares Introduction with reference to Victor Hill and Bach. http://www.maa.org/mathland/mathland_10_14.html

Search MAA Online MAA Home Ivars Peterson's MathLand October 14, 1996 More than Magic Squares "In my younger days, having once some leisure (which I still think I might have employed more usefully), I had amused myself in making . . . magic squares." Benjamin Franklin, who made this comment in a letter written more than 200 years ago, was certainly not the first to experience the fascination of magic squares. People have been toying with these number patterns for more than 2,000 years. Typically, a magic square consists of a set of integers arranged in the form of a square so that the sum of the numbers in each row, each column, and each diagonal add up to the same total. If the integers are consecutive numbers from 1 to n ^2, the square is said to be of n th order. Here's an example of a magic square of the fourth order, made up of the first 16 integers. The sum of the numbers in each row, column, and diagonal is 34. There are 880 possible magic squares of the fourth order, not counting reflections or rotations of each pattern. One of the most remarkable of these squares is one that dates back to India in the eleventh or twelfth century. Notice that not only the rows, columns, and diagonals add up to 34 but also the corner 2 x 2 subsquares. And there's more! The four corner numbers add up to 34, as do the four numbers in the center. Other subsquares (such as 3 + 10 + 6 + 15) give the same result. It's also possible to find "split" subsquares and "split" diagonals that work: 7 + 2 + 14 + 11, and so on. In fact, there is an astonishing number of different ways to get the sum 34 out of this particular magic square.

62. Multimagic Squares Squares that remain magic after entries are raised to various powers. Examples, constructions, bibliography and links compiled by Christian Boyer. (English/French/German) http://www.multimagie.com/

63. Backtracking: 5 X 5 Magic Squares Interactive 5x5 magic square generator using backtracking algorithm. The user can move the numbers manually, watch the computer slowly creating a magic http://www.faust.fr.bw.schule.de/mhb/backtrack/mag5en.htm

tabeinf(laenge,laenge,1,leer,'red',' center') tabeinf(laenge,laenge,2,panu,'green','center ') tabeinf(laenge,laenge,3,leer,'blue','center') mouse show quick X delay (ms) solution No.: Stop at next solution: panmagic bordered panmagic - symmetric symmetric 5 x 5 magic squares requires JavaScript *) A mouseclick on any number of the green-bordered square ("number-pool") moves this number into the first empty field of the red-bordered square ("magic area"); a click on any number of the magic area brings the number back to its original place in the pool. Aim A magic square of order 5 is to be constructed, depending on choice a panmagic and/or a symmetric or a bordered one. When "show" or "quick" is activated, a backtracking algorithm will continue the search for a solution; interruption can be caused by clicking the option "mouse". When a solution is found, the algorithm will stop, and in that case, a search for a next solution may be continued with the corresponding button. The blue-bordered square always contains the last found solution. The delay (in milliseconds) regulates the frequency of image output, if "show" is activated.

64. Magic Squares Of Order 4 Interactive 4x4 magic square generator using backtracking algorithm. The user can move the numbers manually, watch the computer slowly creating a magic http://www.faust.fr.bw.schule.de/mhb/backtrack/mag4en.htm

tabeinf(laenge,laenge,1,leer,'red',' center') tabeinf(laenge,laenge,2,panu,'green','center ') tabeinf(laenge,laenge,3,leer,'blue','center') mouse show quick X delay (ms) Solution #: Stop at next solution: general centralsymm. panmagic mirrorsymm. magic squares of order 4 requires JavaScript *) A mouseclick on any number of the green-bordered square ("number-pool") moves this number into the first empty field of the red-bordered square ("magic area"); a click on any number of the magic area brings the number back to its original place in the pool. Aim A magic square of size 4 x 4 is to be constructed, depending on choice a general, a panmagic or a symmetric one. When "show" or "quick" is activated, a backtracking algorithm will continue the search for a solution; interruption can be caused by clicking the option "mouse". If the Stop-option is activated, the algorithm will stop, whenever a solution is found, and in that case, a search for a next solution may be continued with the corresponding button. The blue bordered square always displays the last found solution. The delay (in milliseconds) regulates the frequency of image output, if "show" is activated. H.B. Meyer

65. Java Applet A documented Java applet that generates magic squares using a given starting number, increment, and starting location. Row, column and diagonal sums are displayed on request. Odd orders up to 7 are accepted. JRE required http://www.ouh.nl/open/eyn/applets/magicsq.htm

66. Interactive Magic Square 3x3 Hope you like this 3 x 3 magic square!! Find out more about magic squares. View Benjamin Franklin s 8x8 magic square in animation 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

67. Magic Square -- From MathWorld magic square Wikipedia, the free encyclopediaA normal magic square contains the integers from 1 to n². magic squares exist for all orders n = 1 except n = 2, although the case n = 1 is trivial—it http://www.astro.virginia.edu/~eww6n/math/MagicSquare.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Recreational Mathematics Magic Figures Magic Squares ... Matrix Types Magic Square A magic square is a square array of numbers consisting of the distinct positive integers arranged such that the sum of the 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 th order general magic square starting with an integer and with entries in an increasing arithmetic series with difference between terms is (Hunter and Madachy 1975).

68. Merlin's Magic Squares a magic squares game with a Linear algebra Background. http://www.cut-the-knot.org/ctk/Merlin.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Cut The Knot! An interactive column using Java applets by Alex Bogomolny Merlin's Magic Squares April 1998 In Merlin's Magic Square , an article that appeared in The American Mathematical Monthly in 1987, Don Pelletier explored the mathematical apparatus behind a toy game known as MERLIN . The game is not quite trivial and the mathematics is simple enough to provide an entertaining exercise in a Linear Algebra class. The original game is played on a 3x3 array of buttons that toggle between two states. The goal of the game is to achieve a target configuration of button states by pressing the buttons. The difficulty lies in that pressing a button alters its state but also toggles states of some neighboring buttons. The applet below generalizes the game in three ways: Buttons are multistate, The target configuration is modifiable, The manner in which pressing a button affects its neighbors is selectable. The applet consists of two 3x3 arrays. On the left, the small one shows the target configuration. To modify the target configuration, click on the squares you want modified. On the right, a bigger one holds the puzzle itself and, if the

69. Magic Squares Game A game with a Linear Algebra background. http://www.cut-the-knot.org/game_st.shtml

This site uses frames, tables and JavaScript. Please get a more intelligent browser. will do.

70. Magic Squares Applet Please Note This applet, which is an update of the original magic squares Applet, has been having problems running on the following system configurations http://www.mste.uiuc.edu/java/michael/magicsquares/

Magic Squares Applet Having trouble running this applet? Download the lastest version of the Java Run Time Environment Click the 'Get It Now' link in the top right corner of the page. The Source Files.... MagPuz.java MagCan.java MagControls.java ColorsWindow.java Credits: 1996: Mike Morton wrote this applet Pavel Safronov updated and recompiled this applet. Michael McKelvey updated and recompiled this applet. Please Note: This applet, which is an update of the original Magic Squares Applet , has been having problems running on the following system configurations: Windows 98 with Internet Explorer 6.0 We have, however, seen it working on the following configurations: Mac with both Safari and IE 5.2.3 Windows 98 with IE Windows Me with IE 6.0 (Java Plug-in 1.4.1) Windows Me with NN 6.1 (Java Plug-in 1.4.1) Windows 2000 with IE Windows XP with IE 6.0 (Java Plug-in 1.4.2) An interesting property discovered by some users is that when the updated version does not run, the original will, and vice versa. If you have any questions/suggestions, send an email to:

71. SquareMaker To Design A Magic Square Or A Sudoku pivari.com SquareMaker a simple tool to create magic square png. Introduce your magic square (every number separated by ,). 5,31,35,60,57,34,8,30, http://www.pivari.com/squaremaker.html

News Progetti Articoli Tools ... Contatti Design your Magic Square (or a Sudoku) by Fabrizio Pivari http://www.pivari.com/ Introduce your Magic Square or Sudoku (every number separated by ,) png jpeg show download Do you like this service? Would you like the same service in your intranet or in your web site? Do you want a client Windows tool with the same features? Contact us News Progetti Articoli ... Contatti

72. Magic Squares A magic square is a rectangular array of numbers, usually from 1 to n2, One of the most famous illustrations of a magic square is in the famous http://www.pballew.net/magsquar.html

Magic Squares    A "magic square" is a rectangular array of numbers, usually from 1 to n so that each column, row, and both diagonals have the same sum.  Other types of magic squares, with other shapes or special properties are common in recreational math.  You can find a very great number of examples of different squares and special features at the web site of Harvey Heinz with additional information on magic stars.   The history of magic squares dates back to at least 1000 BC in China.  A Chinese book called Lo Shu (book of the River Lo) relates the story of how a magic square on the back of a turtle saved the city.  By the 2nd Century BC there were 4x4 magic squres appearing, often in connection with religious practice.  The Islamic/Arab mathematicians probably were introduced to the magic square from India, but they quickly developed squares of higher order.  One of the most famous illustrations of a magic square is in the famous Albrecht Durer woodcut, Melancholia.  The illustration and a blow up of the square can be seen at these links to the St Andrews University site.  Durer was a major contributor to the mathematics of art and is often credited with being the founder of descriptive geometry, yet he is probably known to most students, if at all, for this single woodcut. There are 880 different solutions to the 4x4 magic square, including the one in the Durer painting.

73. The JavaScript Source: 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 0 to make them trade 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! Compare prices Televisions (TVs) Sony Samsung Sharp ... Samsung Compare Prices: 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! White Paper: Information Management Trends In 2005 What's your current data focus? While some companies are still focused on storage and access, others are leveraging their information to be more agile, gain new insights into business performance and make better decisions. Get this white paper and learn how your peers are implementing their data management solutions.

74. Magic Square http://www.smilemathematics.co.uk/flash/magicsq.html

75. Magic Squares This method gives only 1 magic square for any N. There are many others. Methods of creating even magic squares are known, but are not this elegant. http://uzweb.uz.ac.zw/science/maths/zimaths/magic.htm

Creating magic squares Have you ever given someone a problem which they can't solve after hours, when you can produce an answer in a matter of minutes? If not, here's your chance! A magic square is a N x N array of cells, each cell containing one of the first N positive integers, such that the sum of the numbers along any row, column and the two diagonals is a constant. De la Loubere was the French Ambassador to Thailand in 1687-1688 when he learnt the following method for odd values of N. Write 1 in the middle cell of the top row. Now keep writing 2,3,4,... in the up-right direction. If you reach the end of the square, `come out' on its opposite side. If you have to write on a used square, go down one cell. For example, here is a 7 by 7 magic square. This method gives only 1 magic square for any N. There are many others. Methods of creating even magic squares are known, but are not this elegant. Now, go and ask your best friend to create a 99 by 99 magic square... Back to Zimaths Issue 1.4 to Geocities

76. Magic Squares - Mathematics And The Liberal Arts (3) magic squares may first occur in the lo shu diagram, which is often linked The author discusses how the idea of magic squares may have entered the http://math.truman.edu/~thammond/history/MagicSquares.html

Magic Squares - Mathematics and the Liberal Arts To expand search, see Arithmetic and Mathematics and Mysticism . For material on related topics, see The Square . Laterally related topics: Number Systems Numerology Bookkeeping Modular Arithmetic ... Astrology , and Pseudoscience The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews , published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet Biggs, N. L. The roots of combinatorics.

77. TeacherSource . Math . Magic Squares And Stars | PBS Themebased units for K-12 math teachers, including classroom activities and career connections showing how math is used in everyday life. http://www.pbs.org/teachersource/mathline/concepts/historyandmathematics/activit

search options Friday, September 16, 2005 Activity II: Magic Squares and Stars (Grade Levels: 4-8) About Math Concepts Proving the Pythagorean Theorem Magic Squares and Stars The Tower of Hanoi ... More Math Concepts Standards: Standard 1: Number and Operations Standard 9: Connections Objectives: The activity is about the history and creation of a mathematical puzzle or recreation called magic squares. Students will have the opportunity to: use number sense and arithmetic facts to find missing numbers in a magicsquare and/or magic star. follow directions given in words and diagrams to create a magic square of odd order. Student Activity (PDF File) Answers (PDF File) Activity 2 This elementary activity can be used at higher levels by having students determine all possible combinations of 4 digits between 1 and 12 inclusive whose sum is 26 (there are 32) and then creating as many magic stars as they can using their list. Magic Squares and Stars For centuries, mathematicians and individuals interested in recreational mathematics have been interested in magic squares. Practically all historians agree that the magic square had is origin in China centuries before the birth of Christ. The exact origin has been lost, but Oriental tradition holds that the Emperor Yu (c.2200 B.C.) was standing on the bank of the Yellow river when a tortoise appeared with a mystic symbol on its back. This figure came to be know as the

78. Solution For /arithmetic/magic.squares Solution to the /arithmetic/magic.squares problem. These are called magic squares. A magic square of order n (integers from 1 to n*n) has only one possible http://rec-puzzles.org/sol.pl/arithmetic/magic.squares

Solution to the /arithmetic/magic.squares problem These are called magic squares. A magic square of order n (integers from 1 to n*n) has only one possible sum: (n*n+1)*n/2. Odd and even order squares must be constructed by different approaches. For odd orders, the most common algorithm is a recursive scheme devised by de la Loubere about 300 years ago. For even orders, one procedure is the Devedec algorithm, which treats even orders not divisible by 4 slightly differently from those which are divisible by 4 (doubly even). For squares with odd-length sides, the following algorithm builds a magic square: Put 1 in the middle box in the upper row. From then on, if it's possible to put the next number one box diagonally up and to the right (wrapping around if the edge of the grid is reached), do so, otherwise, put it directly below the last one. ...or even See archive entry knight.tour for magic squares that are knight's tours. To get a 4x4 square, write the numbers in order across each row, filling the square... then use the following pattern as a mask: . X X . X . . X X . . X . X X .

79. LoneWolf's Lair: Magic Squares A magic square is an NxN grid that contains all of the numbers from 1 to N2 This application will generate the magic square for the dimension that you 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]

80. Magic Squares magic squares may be perhaps the only area of recreational mathematics to which The classic form of a magic square is a square containing consecutive http://members.shaw.ca/quadibloc/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

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