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A Magic Square of order n is an arrangement of the numbers from 1 to n^2 (n-squared) in an n by n matrix. The sum of any row, any column, or any main diagonal must be the same. The smallest non-trivial case is of order 3.

The same idea can be extended to other shapes such as stars, cubes, circles, and so on. These are called Magic objects.

Games and Puzzles

• A Very, Very Magic Square A 25x25 magic square with 25 sub-squares and other properties. Maple program and PostScript files available for download.
• All You Wanted to Know About Magic Squares Introduction to magic squares. Basic/C computer programs to generate odd squares. Calculates 8x8 squares with any given date on the first line.
• Allmath.com Interactive game to construct Magic Squares. Different degrees of difficulty and size. Hints and solutions provided.
• Allmath.com: Magic Squares Game Interactive game to construct Magic Squares. Different degrees of difficulty and size of square. Hints and solutions provided.
• Amof: Info on Magic Squares The Amazing Mathematical Object Factory generator makes up to 10 squares of order 1 to 10 with just a few clicks. Brief history and introduction included.
• Finding Magic Squares Using CCM The Chemical Casting Model (CCM) generation method is demonstrated with a Java Applet. Source code and description of algorithm included. (English/Japanese) [JRE required for Applet only].
• Generalized Magic Objects Aale de Winkel's site on generalized objects, with generating function f() and conditions g(), sequences and pseudo-rings.
• Grog's Magic Squares Presents the "Magic Carpet" approach by "Grog". Downloads available for analyzers and creators in Excel format.
• 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]
• MAA: Ivars Peterson's MathLand: More than Magic Squares Introduction with reference to Victor Hill and Bach.
• Magic Cubes The site presents a program for finding all possible order 3 cubes, executable and source included.
• Magic Cubes and Tesseracts Examples, original works, algorithms, and theorems on magic cubes, tesseracts, and hypercubes. (English/Japanese)
• Magic Rectangles and Squares Presents 2xn and 3xn rectangles, as well as squares of order 4,5 and 8. Generation of the last by knight's tour.
• Magic Square Puzzle Solve this 4x4 jig-saw puzzle. [Flash]
• Magic Squares Description, properties and on-line construction. Tour of famous squares. (English/German) [Flash optional]
• Magic Squares An introduction, with algorithms for generating odd, and doubly even (4k) squares. References.
• Magic Squares Java applet generates odd ordered squares, with options to transformations by rotations and reflections. Souce code provided.
• Magic Squares and Hyper Cubes Includes Java Applet and source code of square and hyper cube generator. Options for dimensions and offset.
• Magic Squares at Math Cats: Fun, Easy, and Interactive! Introduction to the Durer magic square. Activities including the magic carpet.
• Magic Squares, Magic Stars and Other Patterns Pages on magic squares, stars (original material) and miscellaneous number patterns. Downloads include BASIC programs, Excel and .zip files.
• Making magic squares The Durer and Loh Shu squares. Starting from basic operations, mathematical derive distinct squares from existing ones.
• Math Forum: Magic Squares Applet Java applet and source files.[JRE to run]
• Math Forum: Magic Stars Introduction to Magic Stars. Construction, analysis and transformations.
• Math Forum: Multiplying Magic Squares Includes classroom material covering introduction, the math behind, and methods of construction.
• Math Forum: Suzanne Alejandre: Magic Squares Curriculum resources, student activities and explanations of puzzles.
• Mathemusements Tutorial on how to play magic square games. Gives historical background and different types of illustrations.
• Mathpages: Franklin's Magic Squares How Benjamin Franklin constructed his magic squares. Autobiographical extract included.
• Mathpages: Solving Magic Squares An attempt to solve for all possible magic squares, a mathematical treatment.
• MathWorld: Magic Squares Definition, examples, and construction. Includes algorithms for odd, doubly and singly even squares.
• MathWorld: Panmagic Square Demonstration of the properties of a pan-magic square.
• 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)
• Nested Magic Squares Definition of nested squares. Generation of even- and odd-sided squares, algorithms and matlab function. Samples of order up to 99 are presented.
• Prime Magic Squares Introduction. Prime magic squares, programs and examples.
• PrimePuzzle #297. Queens on magic squares Try to put 8 queens on an 8x8 square such that the queens sit on prime numbers. Readers' solutions provided.
• Program on Hyper Magic Cubes Downloadable program displays random hyper cubes of given order and dimension. It displays simple, pan-n-agonal and perfect hyper-cubes and inlaid squares.
• Science News for Kids: MatheMUSEments Description of Durer's square and an alphamagic square.
• Siamese method The Siamese method generates odd squares of any order above one. Explains the mathematics behind the scenes.
• Simple Magic Square Checker Enter numbers and display the magic square in gif or jpg formats.
• Skeleton Magic Squares Demonstrates methods and algorithms for squares and cubes.
• The 3x3 Distinct Prime Magic Squares Page A paper describing the generation of 3x3 Distinct Prime Magic Squares using a computer program.
• The Anti-Magic Square Project Starting from the definition of an Anti-Magic Square (AMS), the article presents the structure and construction methods of the AMS. Programs written in the c-language are available for download.
• The Magic Encyclopedia Encyclopedia with articles relating to magic squares, cubes, hypercubes and other objects
• The Magic Square Record Includes short description of magic squares. Keeps records for largest hand-written squares between 1975 and 1990.
• 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)
• The Zen of Magic Squares, Circles, and Stars Features reviews, information and index of Clifford A. Pickover's book. Publication focuses on historical and cultural attitudes towards the significance of the squares.
• Ulearn Today : article on Magic Squares Introduction. World records up to 1994.
• Visual Magic Squares and Group Orbits I Visual approach. Definition of Euler and Conway squares, with generating algorithm for the latter via Group Orbits. References.
• Walter Trump: Notes on Magic Squares An introduction, including computer estimates of squares of different orders.

 

 

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