Magic Squares, Magic Stars & Other Patterns

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.)

	Downloads	Sept. 12, 2001   MS Word, MS Excel, and program files available for downloading.
	Franklin Squares	Sept. 12, 2001   All about Ben Franklin's magic squares and magic circle. Included is  a variation of his newly discovered 16x16 and a comparison of his 2 16x16 squares.
	Big Magic Stars	July 15,2001  One solution for each of orders 15a to 20a. Blank graphs for the other patterns in each of these orders. Can you fill in the numbers?
	Order-5 pandiagonals	Mar. 20, 2001  The 36 'essentially different' pandiagonals and how they can be transformed to all 3600 order-5 pandiagonal magic squares.
	New Book	Magic Square Lexicon: Illustrated by H. D. Heinz & J. R. Hendricks It gives you the definitions, limits, examples, illustrations, terminology and everything that you want to know about magic objects. See  cover and details .

Contents

Introduction	A brief introduction to number patterns and these pages.
Magic Squares	The door to a group of pages displaying and talking about magic squares
Magic Stars	I have found the total basic solutions for each order from 5 to 11 and   most for orders 12, 13 and 14. On this page I present some results of this study. Also . . .     A new definition for magic stars.     Examples of all orders from 5 to 14, a total of 30 different patterns (graphs).     All minimum and consecutive solutions for orders 5, 6, 7, 8 prime magic stars     Interesting comparisons between the different orders.
Bibliography	More then 160 books, chapters of books or papers dealing with magic squares, etc.
Glossary	Approximately 125 terms related to magic squares, cubes, tesseracts, stars, etc.
Other Number Patterns	A hodge-podge of miscellaneous patterns from my collection. Included are number patterns, interesting numbers, magic graphs, etc.
All about Credits	My policy  on these pages, about giving credit where credit is due.
Similar web sites	Links to other Magic Square, Magic Star and recreational mathematics web sites.
Site Map	Shows the organization of this site, complete with direct links to over 40 pages.
Talk to me. . .	Suggestions, criticism, share your favorite number patterns, etc. I would really appreciate any comments you wish to make in regard to these pages, number patterns in general and magic stars in particular, or for that matter, any aspect of recreational mathematics.

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		This WWW site is about numbers and the patterns that can be made from them.
		Most patterns shown here should be understandable to anyone with about grade 5 math but will hopefully be of interest also to persons with advanced education..
		The section on Magic Stars is mostly original work & includes material on Orders 5 to 14, a total of 30 patterns.
		This site represents a very small proportion of the material I have in my notes. I hope to be adding to it frequently so please stay tuned.
		The true spirit of delight…is to be found in the mathematics as surely as in poetry.--– Bertrand Russsell (1872-1970)
		To share my love of the subject with others.
		To demonstrate that mathematics can be enjoyed by persons regardless of their education level.
		To further the appreciation of mathematics and the understanding of number relationships.
		Who will enjoy this site? Anyone who loves mathematics in general and numbers in particular. Teachers looking for enrichment material for their math classes.
		Who designed the patterns? Probably most of these patterns have been around for many years and the original author is unknown. Some are of recent origin, and if the author is known has been credited here. And some are of my own original design (at least as far as I know). Most of  the Magic Star patterns and material shown here are my work.
		Who am I? I am a retired bookbinder and have no training in mathematics beyond high school. I have been fascinated by number patterns and have been collecting them since I was a teenager. For the last several years I have been researching magic stars and have made many interesting discoveries. More information about myself and my family are on my Personal Home-page. (Also more Magic Stars & Magic Squares)
		Who are you? Presumably, because you are looking at this site, you too are interested in recreational mathematics. Help explore the mysteries of magic stars. So far, I have been concentrating mostly on the relationships between the different orders. Aside from order-6, very little is known about the features of  the individual orders. If you too find this subject interesting, I would love to hear from you. Tell me about yourself, your special interests, and your favorite number patterns.

 

All about Credits

A Note regarding credits

My policy on all the pages of this WWW site is
If I know of only one source for a pattern, acknowledge it (if possible, obtain permission to use it).
If I have multiple sources for a pattern with no indication of the original source, consider it public domain.

Consequently, if a pattern has no author indicated, I obtained it from multiple locations, or it is my own
composition.
If you see a pattern on these pages that you feel you should receive credit for,   please let me know the details, so I  may reconsider my acknowledgment.

Links to similar web sites

Mutsumi Suzuki's excellent MAGIC SQUARES - a large site, mostly magic squares some magic stars Lots of links
Eric W. Weisstein's Magic Squares - part of Eric's Treasure Trove Of Math, a very comprehensive work
Holger Danielsson's attractive Magic Squares site features online magic square generators and Hendricks work.
Magic Squares by "Grog" - theory of Pandiagonal magic squares. Also a good history of magic squares.
Aale de Winkel – explores theory of magic squares and cubes..Also Pan and 3-D Stars
Aale de Winkel is compiling an Encyclopedia of magic terms, which is also well worth visiting.
Meredith Houlton has a site with information on magic Squares, Cubes and Tesseracts.
Suzanne Alejandre: Magic Squares - this attractive page presents magic squares as a way of teaching math
Concentric Magic Squares – The sum of the cells in each border form an arithmetic progression
Ollerenshaw & Brée’s   Most-Perfect Pandiagonal Magic Squares and book
Fabrizio Pivari's Strange Magic Squares - a different way of looking at magic squares.
F. Poyo has material on magic squares, cubes and hypercubes.
Robert W. Wilke's Nested Magic Squares - simple inlaid Magic squares. Samples. Some theory.
Shin, Kwon Young's Magic Squares - history, methods of solution, samples (up to 30 x 30).
Dave Harper's unusual Magic Square patterns: uses Java script functions to involve the viewer
Anti-Magic Squares by John Cormie & Vaclav Linek  presents an extensive investigation of this subject.
SoftArtes MagicSquares is a magic squares generating program written by Richard W. Bogosian.
Charles Kelly has a Java applet to generate magic squares and hypercubes.
Want to see some BIG! magic squares? Visit Bogdan Golunski's Web page to see the biggest! 10001 x 10001.
Dave Edwards has some nice Java applets for displaying  magic lines of magic cubes, orders 3 to 9.
Marian Trenkler has many PDF or PS files on magic squares, cubes, stars, graphs, etc, that may be downloaded.
Simon Whitechapel has solutions for 1 pattern of each order magic star from 15 to 20. Also articles on other math subjects.
Cambridge University has an interesting site on Secondary School Math Enrichment.
Paul C. Pasles has pages on Franklin and other OLD magic squares.
Mark Swaney has a detailed history of magic squares .
Mark Farrar has a page on magic squares which include several applets and a subscription link to a magicsquares group.

Some other sites on Mathematics

The Math Forum - The premier source on the Web for educational mathematics. All about math, for the student or hobbyist
Frequently Asked Questions in Mathematics
Eric's Treasure Trove of Mathematics - covers a very wide range of material - a big site
Patrick De Geest's World of Numbers- is a very attractive and informative site
Carlos B. Rivera's site, Prime Puzzles and Problems ,  has lots of food for thought
Mike Keith – has lots of good material on interesting numbers
Keven Brown - has a great site on a variety on mathematic subjects.
Harry J.Smith's Fun With Mathematics - a variety of programs and subjects on recreational mathematics
Dr. Michael Ecker's Recreational and Educational Computing  newsletter about recreational mathematics and computing
Ed Pegg Jr.'s http://www.mathpuzzle.com/ has a great sellection of puzzles and lots of recreational math links
Alexander Bogomolny's cut-the-knot large site deals with many math subjects and has almost 2000 links
Monty Phister's site at http://gnarlymath.com/ contains puzzles, games, stories, etc to promote enjoyment of mathematics
For a list of all PPDI's in bases 2 to 10, visit Dr. Lionel E. Deimel's site
Chris Caldwell has a large site dealing with Prime numbers as http://primes.utm.edu/research/primes/
G. L. Honaker, Jr 's & Chris Caldwell's Prime Curios page has many interesting and fascinating facts about individual prime numbers
A great mathematics site for students is  http://www.aaamath.com/
Charles Ashbasher's site has information about the Journal of Recreational Mathematics.
Christian Boyer has constructed a tetramagic square and a pentamagic square (his page is in French).
Reg Brooks has a page on Patterns in Numbers which includes the geometry of DNA.
The World of Trotter Math parades the work of dedicated mathematics teacher Terry Trotter.
Shyam Sunder Gupta discusses RARE and EPORNS numbers (and other numbers).
Walter Schneider has an informative site dealing with digit-related-numbers.

Site Map

Magic Star, Squares, & Other Number Patterns
      Magic Stars
		Magic Star Examples
                        Magic Star Examples - 2 
                        Big Magic Stars
		Magic Star Definitions
		Order-5 Magic Stars
		Order-6 Magic Stars
			Order-6 Solution List
			H. E. Dudeney Features
		Order-7 Magic Stars
                Order-8 Magic Stars
                Order-9 Magic Stars
                Order-10 Magic Stars
                Order-11 Magic Stars
                Prime Number Magic Stars
                        Prime Number Magic Stars - 2 
                Unusual Magic Stars
                Isomorphic-like magic Stars                
                3-D Magic Stars
                Trenkler Stars
      Magic Squares
                A Deluxe Magic Square
                More Magic Squares
                Material from REC
                Unusual Magic Squares
                John Hendricks - Cubes
                Prime Number Magic Squares
                Quadrant Magic Squares
                       Order-13 Quadrant Magic Squares
                       Order-17 Quadrant Magic Squares
                Type 2 Order-3 Magic Squares
                Anti-magic Squares
                Self-similar Magic Squares
                Most-perfect Magic Squares
                Magic square Models
                Transformations & Patterns
                       More Order-4 Transformations
                       Transformations Summary
                       Fellows Transformations
                Order-4 Lists
                       Order4 #1 to 200
                       Order4 #201 to 400
                       Order4 #401 to 600
                       Order4 #601 to 880
                Pandiagonal 5x5
                Franklin squares  
      Number Patterns
                Narcissistic Numbers
                Prime Patterns
                More Prime Patterns
      Glossary
      Bibliography
      Book for Sale
      Downloads