81. Enumeration Of Magic Squares There are 38102400 regular pandiagonal magic squares of order 7. D8 ist greater than C8 because each associative magic square of order 8 can be http://www.trump.de/magic-squares/howmany.html

How many magic squares are there? Results of historical and computer enumeration Order semi-magic (A) normal (B) associative (C) pandiagonal (D) ultramagic (E) Variations of a square by means of rotations and reflections are not counted. Statistical notation: means, that the (unknown) correct number is in the interval with a probability of Ultramagic squares are associative (centrally symmetrical) and pandiagonal. = 1, the so-called Lo Shu is unique. was found by the Frenchman Bernard Frénicle de Bessy in 1693. First analytical proof by Kathleen Ollerenshaw and Herman Bondi (1982). and could be found on the former website of Mutsumi Suzuki. was calculated in 1973 by Richard Schroeppel (computer-program), published in January, 1976 in Scientific American. was calculated by myself in march 2000 using a common PC. Suzuki published the result on his website. I could confirm the result by using other methods. is equal to the number of regular panmagic squares. They may be generated using latin squares, as Leonhard Euler pointed out already in the 18th century. and were proved by A.H. Frost (1878) and more elegantly by C. Planck (1919).

82. Perfect Magic Cubes 6 magic squares would be enough to build a smagic cube. But the above mentioned cubes have at most 4 magic squares in surface planes. They are not s-magic. http://www.trump.de/magic-squares/magic-cubes/cubes-1.html

The Successful Search for the Smallest Perfect Magic Cube by Walter Trump Abstract Since 1866 the smallest known 'perfect' magic cube was of order 7. It was constructed by Andrew H. Frost Most magic cube experts thought that 'perfect' magic cubes of smaller orders do not exist. In September 1st, 2003 I discovered a 'perfect' magic cube of order 6. Together with my friend Christian Boyer from France I searched for a 'perfect' magic order-5 cube and in November 13th, 2003 we finally were successful. History of 'perfect' magic cubes order exists disproved or discovered by date no John Hendricks * no Richard Schroeppel yes Walter Trump Christian Boyer November yes Walter Trump September yes Andrew H. Frost yes Gustavus Frankenstein Single Solutions were found by several authors yes General solution by Mitsutoshi Nakamura July * probably the first disproof was already done before Definitions A 'perfect' magic cube There are 3m +6m+4 straight lines: m rows, m colums, m pillars, 6m diagonals (= short diagonals) and 4 triagonals (= long diagonals). In a (simple) magic cube the sums of the numbers in the diagonals may differ from S.

83. Magic Squares magic squares. of moves. 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 http://www.crews.org/curriculum/ex/compsci/webresources/javascripts/magicsquare/

Magic Squares # of moves: 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.

84. Cynthia Lanius' Lesson: Let's Do MATH - Magic Squares Table of Contents. Checkerboard Squares Dice Roll Get a Five Card Game Tower of Hanoi Shopping Fun magic square Game Fun Math Lessons http://math.rice.edu/~lanius/domath/magicsquare.html

Let's Do Math! Put the numbers in order. (Directions below) Choose Level: Try this mathematical game of skill and logic. The object of this game is to put the numbers in order. Choose a Level (3-EASY to 10-HARD). Press the "Start Game" button to start the game, counter, and timer. Move blocks by clicking on them. A block can be moved only if it is in the same row or column as the hole. You can move multiple blocks by clicking the farthest block that you need to be moved. Order all the numbers in a minimum time with a minimum number of moves...Good luck! Original Games by ©Arun Narayanan, Dec 2002 Free JavaScripts provided by The JavaScript Source Let's Do Math Table of Contents Checkerboard Squares Dice Roll - Get a Five Card Game Tower of Hanoi ... Shopping Fun Magic Square Game Fun Math Lessons Cynthia Lanius

85. Magic Squares And Recursion Solving magic squares using Recursion and Recursive programs. It is very simple to solve magic squares and related problems such as anagrams, permutations, 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

86. MAGIC SQUARES Magic cubes are extensions of some of the principles of magic squares. magic squares are square arrays of numbers, in which the sum of each row column, http://mathsforeurope.digibel.be/magic.htm

MAGIC SQUARES Lies De Sutter An De Brandt Katrien De Bruycker Vicky De Marteau Why this subject ? One day our teacher of Mathematics talked to us about Magic Squares. The magic immediately caught us in it's mysterious arms, so we wanted to know more about this subject. It is merely coincidence or is there another unknown power behind it? Is it the same fascinating magic as we can also find in fractals? Does nature have a secret that it will never reveal totally for mankind? Is the entire universe with it's living creatures build up by a complicated (or simple ?!) pattern? So we decided to start with basic research of these intriguing constructions : Magic Squares. What are they, do they have a use, how were they discovered, what will they learn us in the future ? Let us take you on a journey through this magical world. What Is A Magic Square? A magic square is a simple mathematical game developed during the 1500s. You prepare a single square which is divided into the same number of rows and columns. Then, start filling each little square with the number from 1 to x where x equals the number of rows multiplied by the number of columns. You can only use a number once. Fill each square so that the sum of each row is the same as the sum of each column. In the example shown here, the sum of each row is 34, and the sum of each column is also 34

87. Just Riddles And More Magic Squares a fun collection of puzzles, riddles, quizzes, contests, sweepstakes, challenges, mysteries to solve, tests, games, giveaways, free stuff, shopping, gifts, 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: JADE SHADOW- Unlimited FREE access to all 21 levels of play!  Click Here LIKE THIS GAME? SHARE IT WITH A FRIEND.

88. Nrich.maths.org::Mathematics Enrichment::Magic Squares The Nrich Maths Project Cambridge, England. Mathematics resources for children, parents and teachers to enrich learning. Published on the 1st of each month. http://nrich.maths.org/public/viewer.php?obj_id=1337

89. Nrich.maths.org::Mathematics Enrichment::Magic Squares II The Nrich Maths Project Cambridge, England. Mathematics resources for children, parents and teachers to enrich learning. Published on the 1st of each month. http://nrich.maths.org/public/viewer.php?obj_id=1338&part=index&refpage=topten

90. IOI'96 Day 2 Problem 3: Magic Squares Problem 3 magic squares. Following the success of the magic cube, Mr. Rubik invented its planar version, called magic squares. This is a sheet composed of http://olympiads.win.tue.nl/ioi/ioi96/contest/ioi96m.html

IOI'96 Day 2 Problem 3: Magic Squares Following the success of the magic cube, Mr. Rubik invented its planar version, called magic squares. This is a sheet composed of 8 equal-sized squares (see Figure 3). Figure 3: Initial configuration In this task we consider the version where each square has a different colour. Colours are denoted by the first 8 positive integers (see Figure 3). A sheet configuration is given by the sequence of colours obtained by reading the colours of the squares starting at the upper left corner and going in clockwise direction. For instance, the configuration of Figure 3 is given by the sequence (1,2,3,4,5,6,7,8). This configuration is the initial configuration. Three basic transformations, identified by the letters 'A', 'B' and 'C', can be applied to a sheet: 'A': exchange the top and bottom row, 'B': single right circular shifting of the rectangle, 'C': single clockwise rotation of the middle four squares. All configurations are available using the three basic transformations. A: B: C: Figure 4: Basic transformations The effects of the basic transformations are described in Figure 4.

91. Count On - Kaleidoscope The Lo Shu is a magic square of the order 3. magic squares were thought to have protective powers against illness and evil spirits. http://www.counton.org/magnet/kaleidoscope2/MagicSquare/

According to the ancient Chinese book I Ching (The Book of Changes), in about the year 2800 BC the Emporer Yu was boarding a royal barge to cross the River Shu. He was a tortoise hiding in the reeds. On the back of the animal were some curious markings and the Emporer had a scribe copy them. The result of this, according to tradition, was the discovery of the first magic square, the 'Lo Shu'. It is often called the 'River Writing' because of this story. In this drawing of the 'Lo Shu', numbers are represented by dots - even numbers are black and odd numbers are white. Look at the diagram below, can you see a connection? Try adding up the numbers in the rows, columns and diagonals. The 'Lo Shu' is a magic square of the order 3. Magic squares were thought to have protective powers against illness and evil spirits. From the earliest of times their construction has fascinated mathematicians all around the world. What makes a square magic? The numbers in a magic square sould be the same as the number of cells (for examples in a square of 3 x 3 the numbers are 1 - 9); and they are placed so that each row, column and the two main diagonals all sum to the same total. The smallest magic square is the 'Lo Shu' which is of the order 3. Any other arrangements of order 3 are only reflections and rotations of 'Lo Shu'.

92. Magic Squares Try to put the squares back in order. The 0 repesents the empty spot. Click a square next to the 0 to make them trade places! Get the squares back in order http://www.cybergrace.com/html/magic_squares.html

Here is a fun JavaScript game. Try to put the squares back in order. The repesents the empty spot. 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: Home Knowing God Opportunities Advertise ... Add Your Site

93. Nested Magic Squares A nested magic square is defined as magic square which contains a magic Choosing the proper value of the 3 x 3 magic square which will be nested in a 5 http://members.aol.com/robertw653/magicsqr.html

Nested Magic Squares by Robert C. Wilke 5 x 5 Nested Magic Square Traditionally, a magic square consists of a set of consecutive numbers such that the rows, columns, and diagonals add up to the same number. A nested magic square is defined as magic square which contains a magic square of smaller order. For example, A B C D E F G H I J K L M N O P Q R S T U V W X Y If the above square is magic and nested then the square with the corners G, I, Q, S, will also be magic. Since the nested square has certain properties, then the outer ring, A, B, C, D, E, J, O, T, Y, X, W, V, U, P, K, F can be worked on independently from the inner magic square, greatly reducing the number of calculations required to determine the total number of nested magic 5 x 5 squares. Choosing the proper value of the 3 x 3 magic square which will be nested in a 5 x 5 magic square turns out to be quite easy. The 5 x 5 magic square by definition consists of the numbers 1 through 25, the sum of those numbers equals 325. In selecting the nested 3 x 3 magic square by definition, the smallest magic 3 x 3 square would consist of the numbers 1 to 9, and the largest 3 x 3 magic square would consist of the numbers 17 through 25 because of the consecutive number constraints imposed by the 5 x 5 magic square. It is possible to construct the following chart for the 5 x 5 magic square which only lists whole number solutions to the problem. Whole Number Solutions 5 x 5 Nested Magic Square

94. The Magic Square Everything You Always Wanted to Know about the magic square The New CD! Music to Download. http://www.themagicsquare.com/

Everything You Always Wanted to Know about the Magic Square The New CD! Music to Download Everything You Always Wanted to Know about the Magic Square The New CD! Music to Download

95. Set - How To Make A Magic Square Of Set What you see here is a magic square, much like the addition and subtraction squares you Constructing a magic square may seem complex at first glance, 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:

96. The Original MURDEROUS MATHS Book EXTRA BIT! Making 5x5 magic squares and the Knight s magic square. How to make a 5x5 magic square add up to other numbers. 17, 24, 1, 5, 15. 20, 5, 7, 14, 16 http://www.murderousmaths.co.uk/books/bkmm1x5s.htm

How to make bigger Magic Squares 5x5 Magic Squares 7x7 Magic Squares The 8x8 "Knight's move" Magic Square An Upside Down Magic Square The first MURDEROUS MATHS book tells you all about magic squares, and how to make your own 4x4 square to produce any number. Here are how some bigger squares work. 5x5 Magic Squares This is the basic 5x5 magic square. It uses all the numbers 1-25 and it adds up to 65 in 13 different ways: All 5 horizontal lines add up to 65 All 5 vertical lines add up to 65 The two diagonals both add up to 65 Finally you can add up the four corners and the number in the middle to get 65. How to make a 5x5 magic square add up to other numbers. This square adds up to 62 in 13 ways. You'll see it's very similar to the first square but we've subtracted 3 from each number in a red box. That's why each line adds up to 3 less than 65. If you wanted each line to add up to 80, that's 15 more than 65. So starting with the original square, you'd just add 15 to each number in a red square. However, we can do better than that! How to lay out a 5x5 Magic Square Have another look at the way the numbers are set out in the original square. It uses all the numbers 1-25, and if you follow the numbers round in order you'll see they appear in this pattern:

97. Mudd Math Fun Facts: Making Magic Squares A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Ever wonder how to construct a magic square? http://www.math.hmc.edu/funfacts/ffiles/10001.4-8.shtml

hosted by the Harvey Mudd College Math Department Francis Su Any Easy Medium Advanced Search Tips List All Fun Facts Fun Facts Home About Math Fun Facts ... Other Fun Facts Features 181937199.184678 Fun Facts viewed since 20 July 1999. Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Easy level: Making Magic Squares Figure 1 Figure 2 A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Ever wonder how to construct a magic square? A silly way to make one is to put the same number in every entry of the matrix. So, let's make the problem more interesting- let's demand that we use the consecutive numbers I will show you a method that works when N is odd. As an example, consider a 3x3 magic square, as in Figure 1. Start with the middle entry of the top row. Place a 1 there. Now we'll move consecutively through the other squares and place the numbers 2, 3, 4, etc. It's easy: after placing a number, just remember to always move: 1. diagonally up and to the right when you can

98. Mudd Math Fun Facts: Magic Squares, Indeed! magic squares, indeed! Perhaps you ve seen the magic square This holds for ANY 3x3 magic square (though if the entries contain more than one digit, http://www.math.hmc.edu/funfacts/ffiles/10001.1-5.shtml

hosted by the Harvey Mudd College Math Department Francis Su Any Easy Medium Advanced Search Tips List All Fun Facts Fun Facts Home About Math Fun Facts ... Other Fun Facts Features 181937200.184678 Fun Facts viewed since 20 July 1999. Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Easy level: Magic Squares, indeed! Perhaps you've seen the magic square which has the property that all rows, columns and diagonals sum to 15. Well, it has another "magic" and "square" property! If you read the rows as NUMBERS, forwards and backwards, and square them, then Magic? Presentation Suggestions: If they like this fun fact, ask them to take a minute to see what happens with the columns. Then try the "diagonals" which wrap around the square... The Math Behind the Fact: This holds for ANY 3x3 magic square (though if the entries contain more than one digit, you will have to carry the extra places) using techniques of linear algebra. For instance, for this magic square: you can check that: The reference gives a proof and generalizations.

99. MAGIC SQUARES Examples of magic squares 3 x 3 to 20 x 20 (by TAMORI s method); (Caution! Fundamental set of 4x4 semimagic squares (477 normalized squares) http://www.sinc.sunysb.edu/Class/est56501/Freda Yao Tang/sq1/magic7.htm

Magic Squares Data Base Examples of Magic Squares 3 x 3 to 20 x 20 (by TAMORI's method) (Caution! Errors were found in 4n+2 type squares by Martin Henz) Examples of Magic Squares 3 x 3 to 10 x 10 (by MATLAB) Magic Squares of 4 x 4 (880 = 220 x 4) Complete Magic Squares of 4 x 4 (48 = 3 x 16) Selfcomplement Magic Squares of 4 x 4 (352) ... Fundamental set of Complete Magic Squares of 5 x 5 (144) (by Professor Grogono) Ultra Super Magic Squares of 5 x 5 (Sixteen complete and selfcomplement magic squares) Complete and/or special 8x8 squares 8 x 8 square by a Knight's movement tour (but not magic!) 8 x 8 complete magic square can not be generated by Knight's movement tour 3 x 3 Magic Squares of Prime Numbers 3 x 3 Magic Squares of Prime Numbers ( arithmetic series ) 3 x 3 Magic Squares of Prime Numbers (arithmetic and/or consecutive series ) ... Complete but Irregular Magic Squares (Abe Gakuho's study) Biggest Magic Square Ever (Ralf Laue's web page on the World Records) Fundamental set of 4x4 semi-magic squares (477 normalized squares) Algebraic form of magic squares AMBi Magic Square of 3 x 3 (by Lee Sallows; Additive for row and column, multipl for diagonals)

100. Magic Square Tens are put in the top row, and ones are put back on the magic square. Each child needs a magic square board, 5 pennies, 1 sheet of paper, 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.

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