21. Requirements Management: Magic Square Apr 5, 2004 THE MAGIC SQUARE. Three useful levels of refinement are the following The product idea is the most abstract concept of what the product http://www.jiludwig.com/Magic_Square.html

22. Magic Square - LoveToKnow 1911 MAGIC SQUARE, a square divided into equal squares, like a chessboard, in each of which is placed one of a series of consecutive numbers from i up to the square of the number http://www.1911encyclopedia.org/Magic_Square

Magic Square From LoveToKnow 1911 MAGIC SQUARE, a square divided into equal squares, like a chess -board, in each of which is placed one of a series of consecutive numbers from i up to the square of the number of cells in a side, in such a manner that the sum of the numbers in each row or column and in each diagonal is constant. 4 o rob zrr zoo 3 o x 3 z 6 i 6 From a very early period these squares engaged the attention of mathematicians, especially such as possessed a love of the marvellous, or sought to win for themselves a superstitious regard. They were then supposed to possess magical properties, and were worn, as in India at the present day, engraven in metal or stone, as amulets or talismans. According to the old astro logers, relations subsisted between these squares and the planets. In later times such squares ranked only as mathematical curiosities; till at last their mode of construction was systematically investigated. The earliest known writer on the subject was Emanuel Moscopulus, a Greek (4th or 5th century). Bernard Frenicle de Bessy constructed magic squares such that if one or more of the encircling bands of numbers be taken away the remaining central squares are still magical. Subsequently Poignard constructed squares with numbers in arithmetical pro FIG. I.

23. Magic Square Summary And Analysis Summary | BookRags.com Magic square summary with 23 pages of lesson plans, quotes, chapter summaries, analysis, encyclopedia entries, essays, research information, and more. http://www.bookrags.com/Magic_square

24. "while" Processing Last Line Of Input Twice - C And C++ | Dream.In.Code Oct 4, 2006 class magic_square { private int Size; // Number of rows and columns in the Magic Square array int MagSq55; // Magic Square matrix int http://www.dreamincode.net/forums/topic/12625-while-processing-last-line-of-inpu

25. Magic Square Summary | BookRags.com Magic square. Magic square summary with 19 pages of encyclopedia entries, research information, and more. http://www.bookrags.com/wiki/Magic_square

26. Magic Square Software Downloads, Reviews: Magic Square, Puzzle Games, Magic Line Download Magic Square shareware, freeware, demo, software, files A cool application which lets you play Magic Square right on your computer. http://www.fileguru.com/downloads/magic_square

Home New Software Top Software Top Search ... Contact Categories Business Desktop Internet Development Multimedia Games Utilities Categories Business Desktop Development Education ... Web Development New Software Baby Album DIY Photo Marking Tool WindowsWAMP DarkWave Studio ... ANSMTP SMTP Component Top Software English To Hindi Dictionary H-Dictionary Fake Voice RioDVD Region Free Player ... Driver Updater Pro Top Search Padi Divemaster Exam Driver Robot Golden Alwafi Dictionary Download Drm Identifier ... Bitrate Bitrate Related Software The Magic Square Pseudocode Of Magic Square Description Of A Magic Square Anagram Magic Square ... P Square page All Freeware Magic Square Software Magic Square puzzlegamespack.com Download ... Screenshot A cool application which lets you play Magic Square right on your computer. This game is also sometimes called lights on.. Freeware download of Magic Square 1.0, size 324.25 Kb. Puzzle Games 1.0 puzzlegamespack.com Download Screenshot A cool Puzzle Games pack that includes Jigsaw, Sudoku, Magic Cube, Magic Square and Towers of Hanoi puzzles. You can make Jigsaw puzzle from any of your picture and you get unlimited puzzles of all kind mentioned.. Freeware download of Puzzle Games 1.0, size 850.91 Kb. Magic Lines 3.0.3

27. Cooks.com - Recipes - Magic Square Alternate chocolate chips and walnuts over crust. Sprinkle Ingredients 6 ( chips .. crumbs .. milk .. walnuts ) Searches related to magic square http://www.cooks.com/rec/search/0,1-0,magic_square,FF.html

28. Wapedia - Wiki: Magic Square In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in http://wapedia.mobi/en/Magic_square

Wiki: Magic square In recreational mathematics , a magic square of order n is an arrangement of n numbers, usually distinct integers , in a square , such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n . The term "magic square" is also sometimes used to refer to any of various types of word square Normal magic squares exist for all orders n ≥ 1 except n = 2, although the case n = 1 is trivial, consisting of a single cell containing the number 1. The smallest nontrivial case, shown below, is of order 3. The constant sum in every row, column and diagonal is called the magic constant or magic sum, M . The magic constant of a normal magic square depends only on n and has the value For normal magic squares of order n = 3, 4, 5, ..., the magic constants are: 15, 34, 65, 111, 175, 260, ... (sequence in OEIS At Macworld Expo Apple Inc. cofounder Steve Wozniak created a magic square after asking his audience to suggest a random number for the magic constant Contents: 1. History

29. Magic Square Silver Pendant | Little Treasure Chest Lucky Symbols handmade jewelry. Magic Square is one of the oldest amulets known. It can bring to its wearer good fortune in all aspects of life. http://www.jewels-empire.com/lucky_symbol_jewelry/magic_square.html

30. Magic Square Of Squares Work done by Herman Baer on the magic square of squares problem. http://www.private.org.il/magic_square.html

Herman Baer's Magic Square of Squares page By Uri Raz Report concerning magic square of squares (3x3) Jan/99 =============================================== In "Scientific American" VIII.98 Martin Gardner in his quarter century review mentioned the above problem : rows and columns to have the same magic some of distinct squares as well as both the diagonals. Of the latter I wasnt sure in the beginning, but at any rateI prefered to content my self with the so-relaxed version of the problem. On the other hand both solutions to the relaxed problem I succeeded to find have a square at least e , and we can put down an arrangement of equations like this : (1a') 71^2 - 66^2 = 5 * 137 = 685 = 5*137 [2] = (26, 3) = (19,18) 71^2 - 54^2 = 17 * 125 = 2125 = [4] = (46, 3) = (42,19) 71^2 - 51^2 = 20 * 122 = 2440 = 2^3*5*6 [2] = (46,18) = (42,26) 71^2 - 46^2 = 20 * 117 = 2925 = 3^2*5^2*13 [2] = (54, 3) = (51,18) 71^2 - 42^2 = 29 * 113 = 3277 = 29*113 [2] = (54,19) = (51,26) 71^2 - 26^2 = 45 * 97 = 4365 = 3^2*5*97 [2] = (66, 3) = (51,42) 71^2 - 19^2 = 52 * 90 = 4680 = 2^3*3^2*5*13 [2] = (66,18) = (54,12) 71^2 - 18^2 = 53 * 89 = 4717 = 53*89 [2] = (66,19) = (51,46) 71^2 - 3^2 = 68 * 74 = 5032 = 2^3*17*37 [2] = (66,26) = (54,46) (2'a) 79^2 - 74^2 = 5 * 153 = 765 = 3^2*5*17 [2] = (27, 6) = (21,18) 79^2 - 69^2 = 10 * 148 = 1480 = 2^3*5*37 [2] = (38, 6) = (34,18) 79^2 - 66^2 = 13 * 145 = 1885 =

31. Magic Square More formally, a magic square can be defined as an nby-n matrix such that the sum of any row, column or main diagonal yields the same result (the square s http://www.fact-index.com/m/ma/magic_square.html

Main Page See live article Alphabetical index Magic square In mathematics magic squares consist of a number of integers arranged in the form of a square in such a way that the sum of the numbers in every row, column and diagonal are the same. A magic square may have odd or even number of rows and columns. Usually the magic square is filled up by consecutive numbers from one to N where N is the number of rows or columns. A magic square is designated with reference to this. Thus a magic square of order N will have N number of rows and columns and will be filled by numbers ranging from one to N More formally, a magic square can be defined as an n -by- n matrix such that the sum of any row, column or main diagonal yields the same result (the square's magic constant, denoted M n )); if these numbers are 1, 2,..., n There are many ways to construct magic squares, but the standard (and most simple) way is to follow certain configurations / formulas which generate regular patterns. Certain other restrictions can be imposed on magical squares, resulting, for example, in bimagic trimagic and multimagic squares, and there are also other forms displaying similar characteristics, including magic circles, magic polygons, and magic cubes.

32. Ada Programming/Types/array - Wikibooks, Collection Of Open-content Textbooks May 24, 2010 Therefore, magic_square may simply be declared like this magic_square constant Character_Display = ( SATOR , AREPO , TENET , OPERA , http://en.wikibooks.org/wiki/Ada_Programming/Types/array

Ada Programming/Types/array From Wikibooks, the open-content textbooks collection Ada Programming Types Jump to: navigation search Computing Computer Science ... Ada Programming Contents Declaring arrays Basic syntax With known subrange With unknown subrange ... Ada Quality and Style Guide An array is a collection of elements which can be accessed by one or more index values. In Ada any definite type is allowed as element and any discrete type, i.e. Range Modular or Enumeration , can be used as an index. edit Declaring arrays Ada's arrays are quite powerful and so there are quite a few syntax variations, which are presented below. edit Basic syntax The basic form of an Ada array is: array of where Index_Range is a range of values within a discrete index type, and Element_Type is a definite subtype. The array consists of one element of "Element_Type" for each possible value in the given range. If you for example want to count how often a specific letter appears inside a text, you could use: type is array (Character) of Natural; As a general advice, do not use Integer as the index range, since most of the time negative indices do not make sense. It is also a good style when using numeric indices, to define them starting in 1 instead of 0, since it is more intuitive for humans and avoids

33. Magic Square - ENotes.com Reference In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in http://www.enotes.com/topic/Magic_square

34. Lukol.com - Science Math Recreations Magic Square Starting from the definition of an AntiMagic Square (AMS), the article presents the structure and construction methods of the AMS. http://www.lukol.com/Top/Science/Math/Recreations/Magic_Square

Lukol Directory Science Math Recreations ... Math Forum: Multiplying Magic Squares Includes classroom material covering introduction, the math behind, and methods of construction. http://mathforum.org/alejandre/magic.squar... Mathpages: Solving Magic Squares An attempt to solve for all possible magic squares, a mathematical treatment. http://www.mathpages.com/home/kmath295.htm The Perfect Solutions for the Magic Square Brief history and definition. Construction algorithms are presented for the odd, even, and singly even squares. (English/Korean) http://user.chollian.net/~brainstm/MagicSq... Walter Trump: Notes on Magic Squares An introduction, including computer estimates of squares of different orders. http://www.trump.de/magic-squares/index.ht... 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/Magic... 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/

35. Magic Square -- From Wolfram MathWorld A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, , n^2 arranged such that the sum of the n numbers in any horizontal, vertical http://mathworld.wolfram.com/MagicSquare.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Interactive Demonstrations 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 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 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).

36. Allmath.com Magic Squares Game Interactive game to construct Magic Squares. Different degrees of difficulty and size of square. Hints and solutions provided. http://www.allmath.com/magicsquare.php

Online Dictionary allmath.com Allmath Homepage Math Tools Flash Cards!!! Metric Converter Games The Magic Square Reference Pages Math Glossary Math Headlines Multiplication Table Metric Conv. Factors ... Other Math Links Ask The Experts Visit Dr. Math Visit our Dictionary! ALLwords.com About allmath.com Welcome! Who are We? Advertise Contact Us yahooBuzzArticleCategory = "entertainment"; yahooBuzzArticleHeadline = "math magic squares"; http://www.allmath.com/magicsquare.php Magic squares have been a fascinating topic in mathematics for centuries. They are formed by filling in all the squares with the numbers starting from one so that the sum of all row, columns, and diagonals is the same. If you know how to play, you can use the square below. Otherwise, read the instructions and fill in the form at the bottom to begin a new game. Numbers to use: Numbers you have used: In this game, you must fill in the magic squares. In the easy level, you are given the numbers beginning from one and must create a magic square. The medium difficulty level is similar, but instead of getting each row, column, and diagonal to add to the same value, each one has a different value which is given to you. In the hard level, you are given random numbers, some positive and some negative, and you must place them in the square to get the sums given to you. To start playing, you first select whether you want a 3 by 3 square, 4 by 4 square, or 5 by 5 square. As the square gets bigger, the puzzle becomes more difficult. Next, you select the difficulty level you wish to play.

37. Math Forum: Suzanne Alejandre: Magic Squares Curriculum resources, student activities and explanations of puzzles. http://mathforum.org/alejandre/magic.square.html

A Math Forum Web Unit Suzanne Alejandre's Magic Squares Suzanne's Math Lessons Suzanne's Tessellation Lessons Magic squares received their name because there are so many relationships between the sums of the numbers filling the squares. Students often believe that "mathematics" was "written" by one person. In these pages you will find that the magic square mathematical game has existed throughout history and in many different parts of the world. Math is all around us and your mind will see it when you're ready! What is a Magic Square? Allan Adler defines and discusses some special properties of magic squares. Magic Squares - History, Mathematics, Geography Lo Shu Magic Square The Back of the Divine Tortoise Classroom Activity China - the country of Lo Shu ... GEOM ART RY - Ralph Martel Benjamin Franklin's 8x8 Magic Square The World of Benjamin Franklin - the Franklin Institute Classroom Activity Pennsylvania, U.S.A. - Benjamin Franklin's State ... Classroom Activity Where's the Math? - with Allan Adler Classroom activities: multiplying magic squares Constructing magic squares Defining the magic square; special properties ... Review: Squaring a magic square Mike Morton's Magic Square Java Applet Updated Magic Square Java Applet by Pavel Safronov and Michael McKelvey Dubi Kaufmann's Magic Square Puzzle H. B. Meyer's

38. Magic Square Jigsaw Puzzle Solve this 4x4 jigsaw puzzle. Flash http://www.dubster.com/math/

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39. Math Forum: Allan Adler - What Is A Magic Square? Multiplying Magic Squares are techniques developed by Allen Adler and made into Web pages by Sarah Seastone and Suzanne Alejandre. http://mathforum.org/alejandre/magic.square/adler/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

40. Magic Squares At Math Cats: Fun, Easy, And Interactive! Introduction to the Durer magic square. Activities including the magic carpet. 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.

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