61. Solving Magic Squares An attempt to solve for all possible magic squares, a mathematical treatment. http://www.mathpages.com/home/kmath295.htm

Solving Magic Squares Note 1 The Determinants of 4x4 Magic Squares Magic Square of Squares Orthomagic Square of Squares ... Return to MathPages Main Menu

62. Magic Square | Define Magic Square At Dictionary.com –noun a square containing integers arranged in an equal number of rows and columns so that the sum of the integers in any row, column, or diagonal is the same. Use Magic http://dictionary.reference.com/browse/Magic square

63. 4x4 Magic Square Solver Given at least two numbers, provides possible solutions for a 4x4 magic square where the desired sum is 34. http://quasistoic.org/fun/magicsquare/

4x4 Magic Square Solver I vaguely remember hearing about Perplex City when it was first launched, but I was too caught up in just about everything else to take too much notice. I do remember thinking that a worldwide puzzle/scavenger hunt game with an online component sounded right up my alley, but I was disappointed that the "crossover into real-life" events were centered in a country I had never set foot in. A number of times since then, I've been reminded of the game's existence, most recently when I heard that Perplex City would be having its first official U.S. event right here in San Francisco...on a day when I already had obligations. In spite of my inability to attend the event, there's been a resurgence in my interest, and last Friday while in Berkeley for a concert I picked up a few packs of "PC" cards from Games of Berkeley. Whoo boy, the good times are a-startin'. I love myself a good mental workout, and the Perplex City cards provide just that in diverse forms and at varying intensity. From pattern-matching to pop-culture knowledge, logic puzzles, physics problems, political trivia all abound. Probably my favorite aspect of Perplex City problem-solving so far is scripting solutions to some of the more complicated puzzles. When I was working on a solution for card #098 'Magic Square', I came up with a script which can be used to solve any 4 by 4 magic square, where the rows, columns, and diagonals all add to the same number.

64. Magic Squares, Stars, And Other Pages on magic squares, stars with total basic solutions for each order from 5 to 11, and miscellaneous number patterns. http://recmath.org/Magic Squares/

M I hope you enjoy these examples from my collection of number patterns. This site has three sections, with pages on magic squares, magic stars (a lot of original material) and miscellaneous number patterns. This site should be of interest to middle and high school students and teachers, and anyone interested in recreational mathematics. Wherever there is number, there is beauty. Proclus (410-485 A.D.) Moved to magic-squares.net A few pages of this collection were first posted to Geocities in March 1998 as www.geocities.com/CapeCanaveral/Launchpad/4057/. Over the years the collection grew in size and became www.geocities.com/~harveyh/ . On Oct. 14/09 the entire site has been moved as is, to this new location, because Geocities is closing it's doors. Magic Square Update Oct. 2009. 3 new types. Normal squares just a subset. Postage stamp m.s., etc. June 2009 Thanks to Ed Pegg, this geocities site is now mirrored on http://recmath.org/ Ultra-magic Squares A set of magic squares constructed by Walter Trump. Posted July 2008. Magic Tesseracts Nov. 2007 All about magic tesseracts. A new site (because of space) consisting of 11 pages.

65. LogoMation LogoMation is an easy to learn computer programming environment. It is available on Window (95, 98, NT 4.0), and Macintosh. http://www.magicsquare.com/LM2/

LogoMation : A Computer Language for Education If you arrived here looking for the LogoMation Sports Screen Saver , click HERE LogoMation is an easy to learn computer programming environment. It is available on Window (95, 98, NT 4.0, Win2000), and on Macintosh. The included LogoMation book provides step-by-step, clear and easy to follow instructions on using the development environment. LogoMation is for Middle and high-School students, learning computer programming. College-level introductory computer science courses. People who never programmed a computer, and always wanted to try. Seasoned programmers who want to have fun with animation. Please click on the buttons on the left to learn more about LogoMation. LogoMation support: Support@MagicSquare.com Questions and information: info@MagicSquare.com

66. The 3x3 Distinct Prime Magic Squares Page A paper describing the generation of 3x3 Distinct Prime Magic Squares using a computer program. http://www.dcs.gla.ac.uk/~pd/Numbers/MagicSquares/

The 3x3 Distinct Prime Magic Squares Page This page is designed to make available some material on 3x3 distinct prime magic squares, for use by: fans of magic squares and prime numbers, and possibly even serious number theorists. More importantly, this material can be used to introduce the concept of algorithmic complexity to novices - this page will be updated in June to include such material. What is a 3x3 distinct prime magic square? It's exactly what it sounds like: a collection of 9 distinct prime numbers organised into a 3x3 grid in such a way that all of the rows, columns, and diagonals sum to the same value (which is three times the value at the centre of the grid). Why study 3x3 distinct prime magic squares? That's a good question. I wrote the prime number generator long ago as a coding exercise, and the 3x3 distinct prime magic square generator was written to answer a puzzle in the department newsletter. I was surprised at how many such squares there are, so just left the generator running.... it eats cycles quietly on my workstation and happily churns out new squares. Whether anyone else is interested in these squares isn't clear, but I thought I'd put the material up on the web in case anyone wanted to look at them. The algorithmic content is useful in that it makes a nice example of how vastly improved algorithms can follow from a deeper understanding of a problem. If you are aware of an application of these squares in any problem area, or of interesting mathematical observations concerning these squares, please do let me know. I'm not currently aware of any practical use for them whatsoever!

67. Ed's Magic Squares Features magic squares, rectangles and the author s signature magic diamonds commemorating historic and personal milestones. http://www.eds-magic-squares.com

Magic Squares by Edward W. Shineman Jr. Puzzled about magic squares? Check here Magic Squares Magic Rectangles Magic Diamonds ... Contact Ed www.eds-magic-squares.com

68. Magic Square You Shouldn't Enter a Lottery, Contest, or Any Game of Chance Without Touching the Magic Square First! http://www.calastrology.com/magicsquare.html

I'm looking for: You Shouldn't Enter a Lottery, Contest, or Any Game of Chance Without Touching the Magic Square First! From the mystical Hebrew tradition known as the Kabbala comes the most influential amulet of them all, the Magic Square. One of the oldest amulets known to man, the Magic Square could bring a person good luck in all aspects of his or her life. What Is the Magic Square? An extremely powerful talisman containing numbers that add up to the same total whether adding horizontally, vertically, or diagonally. The Magic Square could dramatically sharpen your instincts and increase your luck. How Do You Use It? Merely hold the Magic Square while you're selecting numbers and the correct numbers may begin to emerge from your subconscious mind. The Magic Square may be particularly useful in games of chance, such as the lottery. The Magic Square Could: Help you select lucky numbers for the lottery. Improve your odds for winning Bingo, contests, etc. Improve your luck in all aspects of life. Turn your hard luck into good luck Order the Magic Square today and turn your hard luck into good luck. Plated in 14-karat gold, the Magic Square can be worn or carried.

69. Bordered Magic Squares Includes bordered or consecutively concentric magic squares. Demonstrates odd and even numbered squares and makes bordered squares of any size. http://users.eastlink.ca/~sharrywhite/BorderedMagicSquares.html

Home How Many Border 5 Downloads Bordered Magic Squares Terminology There are differences in the usage of the term Bordered Magic Squares . Some sites use it to mean the same as Concentric Magic Squares while others restrict it to mean only the Consecutively Concentric Magic Squares Most of the squares presented here are of the restricted consecutively concentric variety. These have the property that each nested square consists of consecutive integers. A nested square of order m may be converted to a normal magic square by subtracting the of the border order n square and adding , which is equivalent to subtracting Background The algorithms here were first programmed on the pocket calculators HP-67 and HP-41CV . The bones The programs are designed to compute one cell number at a time, given the row and column. A (row, column) is entered to calculate any cell number or, optionally, the square can be stepped through, computing one cell at a time, by repeatedly pressing the R/S button. Because the squares are not stored in the calculator, the biggest square that can be made is determined by the computational and display capability of the calculator, and, time and patience Odd Order Squares Example Squares: Order 3, 5, 7

70. Magic Squares Magic Squares A magic square is a rectangular array of numbers, usually from 1 to n 2, so that each column, row, and both diagonals have the same sum. 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.�The image at right is from the web page of Karen Verschooren and Tine Uytdenhouwen 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.

71. Magic Squares Applet Java applet and source files.JRE to run http://mathforum.org/alejandre/java/magpuz/MagPuz.html

Magic Squares Applet Note: Mike Morton wrote this applet while participating in the Math Forum's 1996 Summer Institute. Once we cataloged this applet in our Math Tools project, its functionality in different browsers and platforms was raised in a discussion. If this version does not work with your platform and/or browser, try this version of the Magic Squares Applet revised by Pavel Safronov and Michael McKelvey. Find accompanying lessons and student activities here The Source Files.... MagPuz.java MagCan.java MagControls.java ColorsWindow.java If you have any questions/suggestions, please send them to Suzanne Alejandre

72. Magic Square Activity Supplies 3x5 index cards, markers, pencils, rulers, pennies (5 for each child) 1. A magic square is a square array of consecutive natural numbers, 1, 2, 3 http://www.math.nmsu.edu/~breakingaway/Lessons/magicsquare1/magicsquare.html

Magic Square and five pennies Activity Supplies: 3x5 index cards, markers, pencils, rulers, pennies (5 for each child) A magic square is a square array of consecutive natural numbers, 1, 2, 3, ..., such that the sums of the numbers in each row, in each column, and on both diagonals are the same. Creating magic squares is a very ancient art, and the smallest and best known one is Memorizing addition facts, and being able to recall them instantly, is as important now as it ever was. But many methods for teaching addition facts that have been used previously are not acceptable today. One method involving the whole class worked as follows: The teacher says two numbers, for example "seven, six", and points to a student. The student must immediately get up, stand at attention, answer, "thirteen", and sit down. In this way, in a few minutes every student would answer two or three questions, and everyone was paying attention because he or she could be asked to answer the next question. This kind of "military drill" is not acceptable anymore, but achieving almost automatic computation of sums of small numbers requires a large amount of practice distributed over a long period of time, which is quite a boring activity. The magic square board, described below, is designed to provide practice in addition facts which can be done with the whole class under the teacher's direction.

73. Magic Square Game Maths Is Fun is ICRA Registered. Copyright � 2008 MathsIsFun.com. Magic Square Game. Put the pieces together so that the rows and columns add to 30. http://www.mathsisfun.com/games/magic-square-game.html

74. Magic Squares And Hyper Cubes Includes Java Applet and source code of square and hyper cube generator. Options for dimensions and offset. http://net.indra.com/~charliek/

Introduction Magic squares are arrangements of numbers into squares, cubes, or hyper-cubes where the sum of each row and each column is identical. Additionally, the sum along the "principal diagonals" is also equal to the same number. The magic square below was known in antiquity in China, Greece, and Egypt. It is interesting to note that subtracting a constant from each cell in a magic square or hyper-cube does not change its essential property: namely, that the sum of any row, column, or principal diagonal is identical. For example, the following magic square is obtained by subtracting 5 from each cell in the magic square above: An interesting property about this square is that all rows, columns, and principal diagonals sum to 0. Also, it is easy to visually recognize symmetry in the square. This web site contains: Examples Instructions Applications Theory ... Symmetry - examples using "tuples" Generator - program for generating magic squares and Hyper Cubes Other Web Sites for Magic Squares and Cubes Please see the Instructions Page before viewing the Magic Squares and Hyper Cubes Generator Site Map: Postal address P.O. Box 567; Niwot, CO 80544

75. Panmagic Square -- From Wolfram MathWorld Demonstration of the properties of a pan-magic square. http://mathworld.wolfram.com/PanmagicSquare.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Vantieghem Panmagic Square If all the diagonals including those obtained by "wrapping around" the edgesof a magic square sum to the same magic constant , the square is said to be a panmagic square (Kraitchik 1942, pp. 143 and 189-191). (Only the rows, columns, and main diagonals must sum to the same constant for the usual type of magic square.) The terms diabolic square (Gardner 1961, pp. 135-137; Hunter and Madachy 1975, p. 24; Madachy 1979, p. 87), pandiagonal square (Hunter and Madachy 1975, p. 24), and Nasik square (Madachy 1979, p. 87) are sometimes also used. No panmagic squares exist of order 3 or any order for an integer . The Siamese method for generating magic squares produces panmagic squares for orders with ordinary vector (2, 1) and break vector (1, The Lo Shu is not panmagic, but it is an associative magic square . Order four squares can be panmagic or associative , but not both. Order five squares are the smallest which can be both associative and panmagic, and 16 distinct

76. Magic Squares And Dihedral Group D4 Java applet generates odd ordered squares, with options to transformations by rotations and reflections. Source code provided. http://www.wou.edu/~burtonl/magic.html

Magic Squares The sum of the numbers in each row, column and main diagonal is the same. It is called the magic constant . In the applet below, magic squares of order (size) 3, 5, 7 and 9 may be explored. A magic square of odd order n contains the integers from 1 to n , arranged so that the rows, columns and diagonals sum to the magic constant ( n n + 1)) / 2. The middle number in each square is ( n + 1)/2. Read more about the squares and the applet here applet-java must be enabled Applet Controls Reset : reset to initial square R0 : rotate degrees clockwise R1 : rotate 90 degrees clockwise R2 : rotate 180 degrees clockwise R3 : rotate 270 degrees clockwise FH : flip horizontally FV : flip vertically FX+ : flip on the line y = x FX- : flip on the line y = -x Order : select the size of the square The Applet The buttons in the applet apply a set of motions to the square which is known in group theory as the dihedral group D4 . The motions are all rotations and reflections of the square in the Euclidean plane, which are rigid in the sense that they do not bend, tear or stretch the square. These rotations and reflections preserve the magic quality of a magic square.

77. Lo Shu Square - Wikipedia, The Free Encyclopedia Because north is placed at the bottom of maps in China, the 3x3 magic square having number 1 at the bottom and 9 at the top is used in preference to the other rotations/reflections. http://en.wikipedia.org/wiki/Lo_Shu_Square

Lo Shu Square From Wikipedia, the free encyclopedia Jump to: navigation search The 洛書 luòshū. Modern representation of the Lo Shu square as a magic square Lo Shu Square simplified Chinese traditional Chinese pinyin luò shū ; also written 雒書; literally: Luo (River) Book/Scroll) or the Nine Halls Diagram simplified Chinese traditional Chinese pinyin jiǔ gōng tú ), is the unique normal magic square of order three. Lo Shu is part of the legacy of the most ancient Chinese mathematical and divinatory ( Yi Jing 易經) traditions, and is an important emblem in Feng Shui (風水), the art of geomancy concerned with the placement of objects in relation to the flow of qi (氣) 'natural energy'. Chinese legends concerning the pre-historic Emperor Yu (夏禹) tell of the Lo Shu, often in connection with the Ho Tu (河圖) figure and 8 trigrams . In ancient China there was a huge deluge : the people offered sacrifices to the god of one of the flooding rivers, the Lo river (洛水), to try to calm his anger. A magical turtle emerged from the water with the curious and decidedly unnatural (for a turtle shell) Lo Shu pattern on its shell: circular dots giving unitary (base 1) representations ( figurate numbers ) of the integers one through nine are arranged in a three-by-three grid.

78. Making Magic Squares The Durer and Loh Shu squares. Starting from basic operations, mathematical derive distinct squares from existing ones. http://www.netcomuk.co.uk/~jenolive/roymagic.html

Making magic squares Strictly speaking, any magic square should have all these properties except for the last one. So, for example, a 3 x 3 magic square will use the numbers from 1 to 9. I show below a copy of the earliest known magic square, the Chinese Loh-Shu, from about 2800 BC. I've added colour here to make the distinction between the odd and even numbers stand out more clearly. In fact the yellow blobs should be white, being Yang symbols or emblems of heaven, and the red blobs should be black, being Yin symbols or emblems of earth. Now we'll look at 4 x 4 squares in more detail. If we relax the rule about the numbers having to run from 1 to 16 we can vastly increase the possibilities and also see some interesting maths in action. All magic squares obey two fundamental rules. Rule (1) If you multiply every element in a magic square by the same number then the result is also a magic square. Here's an example, showing M M and You can see that the basic structure of the square is maintained and the new magic square has a magic sum of 3 x 34 = 102. Rule (2) If you add two magic squares then the result is also a magic square. I've shown an example of this below, using

79. Magic Encyclopedia Magic Square, Magic Cube, Magic Tesseract, Magic Hypercube, Encyclopedia with articles relating to magic squares, cubes, hypercubes and other objects http://www.magichypercubes.com/Encyclopedia/

mathworld.wolfram.com hypercubes "magic stars squares and other subjects" mathworld.wolfram.com hypercubes "magic stars squares and other subjects" ... "Magic Squares and Hypercubes"

80. Magic Squares An normal magic square is an arrangement of the numbers 1, 2, 3, n2 in a square array, with the property that the sum of every row and column, http://www.jcu.edu/math/vignettes/magicsquares.htm

Vignette 20 Magic Squares An normal magic square is an arrangement of the numbers 1, 2, 3, ... n in a square array, with the property that the sum of every row and column, as well as both diagonals, is the same number. An example of a normal magic square is You can verify that each of the three rows, the three columns, and the two diagonals add to 15. Magic Squares in History Magic squares have been studied for at least three thousand years, the earliest recorded appearance dating to about 2200 BC, in China. In the 9 th magic square with some very interesting properties is attributed to him. What is the Magic Sum? Given an normal magic square, suppose M is the number that each row, column and diagonal must add up to. Then since there are n rows the sum of all the numbers in the magic square must be . But the numbers being added are 1, 2, 3, ... n , and so 1 + 2 + 3 + ... + n . In summation notation, . Using the formula for this sum, we have , and then solving for M gives . Thus, a normal magic square must have its rows, columns and diagonals adding to

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