Magic Squares, Magic Stars & Other Patterns

Magic Squares Home Magic Stars Home Misc. Number Patterns

Welcome.gif (11293 bytes)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
A few pages of this collection were first posted to Geocities in March 1998 as Over the years the collection grew in size and became 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.
Magic Squares, Magic Stars & Other Patterns June 2009 Thanks to Ed Pegg, this geocities site is now mirrored on

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.
Also discusses the relationships between hypercubes of different dimensions.
New pages:  October//08 Knight-tours,   March/09 The Unfolded Tesseract


Magic Cubes

Dec. 2002 Posted 11 pages on magic cubes to a new site. This site has now grown to 45 pages (June/09).

The Magic Square Lexicon : Illustrated is now in it's second print run!
And now a free PDF version of the first one-quarter of the book.

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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.


More then 160 books, chapters of books or papers dealing with magic squares, etc.


Some of my Word, Excel and Basic program files.


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 70 pages.
Also shown are the 45 pages on the related Magic Cube site and
the 13 pages on the related Magic Tesseract site.

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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December, 2007 Note:  This page was originally posted in 1998. Much of this Introduction section is now dated. In recent years my interests have concentrated mostly on Magic hypercubes, resulting in a new site at, and recently the tesseract site mentioned above.

What.gif (3099 bytes) bullet.gif (1129 bytes) This WWW site is about numbers and the patterns that can be made from them.
bullet.gif (1129 bytes) 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..
bullet.gif (1129 bytes) The section on Magic Stars is mostly original work & includes material on Orders 5 to 14, a total of 30 patterns.
bullet.gif (1129 bytes) 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.

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Why.gif (2558 bytes) bullet.gif (1129 bytes) The true spirit of delight…is to be found in the mathematics as surely as in poetry.--– Bertrand Russsell (1872-1970)
bullet.gif (1129 bytes) To share my love of the subject with others.
bullet.gif (1129 bytes) To demonstrate that mathematics can be enjoyed by persons regardless of their education level.
bullet.gif (1129 bytes) To further the appreciation of mathematics and the understanding of number relationships.

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Who.gif (2844 bytes) bullet.gif (1129 bytes) Who will enjoy this site?
Anyone who loves mathematics in general and numbers in particular.
Teachers looking for enrichment material for their math classes.
bullet.gif (1129 bytes) 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.
bullet.gif (1129 bytes) 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)
bullet.gif (1129 bytes) 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.

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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.

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Links to similar web sites

John R. Hendricks (1929 - 2007) still has a website showing  Perfect & Inlaid magic tesseracts, as well as other math topics.
Aale de Winkel is compiling an Encyclopedia of magic terms, which is well worth visiting.
Christian Boyer has constructed  tetramagic and pentamagic squares. His site has much information on multimagic squares and cubes.
Mutsumi Suzuki's excellent MAGIC SQUARES is now back - hosted by  MathForum
Dwane Campbell has a comprehensive site on Perfect (Nasik, pan-2-n-agonal) magic hypercubes. He includes generator programs for these forms.
Walter Trump has a several sites dealing with all aspects of magic hypercubes. He also discusses counting.
Abhinav Soni has a new site on magic cubes that features a hyper magic cube generator.
Mitsutoshi Nakamura also has a lot of material and original research on his cubes site. Also, an excellent page on hypercube definitions.
Kanji Setsuda has a large site and much information on magic squares. His English pages are here.
Eric W. Weisstein's Magic Squares - part of Eric's Treasure Trove Of Math, a very comprehensive work
Francis Gaspalou has an excellent site dealing with methods and tools for enumeration of magic squares.
Magic Squares by "Grog" - theory of Pandiagonal magic squares. Also a good history of magic squares.
Suzanne Alejandre: Magic Squares - this attractive page presents magic squares as a way of teaching math
Robert W. Wilke's Nested Magic Squares - simple inlaid Magic squares. Samples. Some theory.
Anti-Magic Squares by John Cormie & Vaclav Linek  presents an extensive investigation of this subject.
saMagicSquares is a magic squares generating program written by Richard W. Bogosian.
Charles Kelly has a Java applet to generate magic squares and hypercubes.
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's  page on magic squares includes several applets and a subscription link to a magic squares group.
To see what can be done with art and magic squares, visit Paul Heimbach's Web site.
G. D. Mutch's Natural Matrix Law is a large site which discusses practical (?)  applications of magic squares.
Edward Gutierrez has a new site where he squares similar to associated, but with center cell containing 1/2(n2-1)
Harry White has a new site (March 2009) on Bordered Magic Squares. He has computed the number of these up to order-14.
Magic Squares Newsgroup at

Magic Cubes Newsgroup at
For magic Squares of Luo Shu format, see Dr. Robert Dickter's
Arie Breedijk has a useful site containing many methods of magic square construction.

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
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 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 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
G. L. Honaker, Jr 's & Chris Caldwell's Prime Curios page has many fascinating facts about individual prime #
A great mathematics site for students is
Charles Ashbasher's site has information about the Journal of Recreational Mathematics.
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.
Charles-. Jean has a good Dictionary of recreational mathematics (in French).
A prime spiral arranged as a circle is thoroughly discussed and illustrated at Rom Sacks
Math for Kids at is an entry point to many web sites with interesting mathematical content.
David Singmaster's source lists on Recreational Mathematics at

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Site Map

Magic Star, Squares, & Other Number Patterns  (Geocities and Recmath servers)
      Magic Squares
                A Deluxe Magic Square
                More Magic Squares
                More Magic Squares-2
                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
                       Most-perfect Bent diagonal 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  
                Multimagic Squares
                Perimeter Magic Triangles
                Perimeter Magic Polygons >k=3
                Magic 3-D Polygons & Graphs
                Magic Knight Tours
                Compact-Magic Squares
                Ultra-magic Squares
                Magic square update
      Magic Stars
		Magic Star Definitions
                Magic Star Examples
                        Magic Star Examples - 2 
                        Big Magic Stars
		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
                Magic 9x5 Hexagrams                
                3-D Magic Stars
                Trenkler Stars
                Magic Star Puzzles
                Star Updates
                        Simon Whitechapel
                        Jon Wharf
                        Andrew Howroyd
      Number Patterns
                Prime Patterns
                More Prime Patterns
                Narcissistic Numbers
      Book for Sale
Magic Cubes - Introduction  (Shaw server- but connected by Button Bar)
      Magic Cubes - The Basics
      Magic Cube - Definitions
      Perfect Magic Hypercubes
      Magic Cubes - the Road to Perfect
      The Early Cubes
      A. H. Frost's Magic Cubes
      F. A. P. Barnard's Magic Cubes
      Magic Cubes - Order 3
      Magic Cubes - Order 4
      Magic Cubes - Order 5
      Magic Cubes - Order 6
      Magic Cubes - Order 7
      Magic Cubes - Order 8
      Magic Cubes - Order 9
      Magic Cubes - Order 10
      Magic Cubes - Order 11
      Magic Cubes - Order 12
      Magic Cubes - Order 13
      Arnoux's Perfect Cube
      Arnoux Patterns
      Modulo Magic Cubes
      Multimagic Cubes
              Monster Cubes
              Boyer's Bimagic 16 Cube
              Boyer's Bimagic 32 Cube
      Magic Cube Groups
      Prime Number Magic Cubes
      Magic Cubes - Multiply
      Composition Magic Cubes
      Hendricks Inlaid Magic Cubes
      The Heinz X6 Magic Cube
      Self-similar Magic Cubes
      Magic Cubes - Pan and Semi-pan
      Unusual Magic Cubes
      Most-perfect cubes
             Order 16 perfect cubes
      Magic Cube Bibliography
      Cube Downloads
      Summary of this Magic Cube Site
      Cube Update-1
      Cube Update-2
      Cube Update-3
      Cube Update-4
Magic Tesseracts - Introduction  (Shaw server- but connected by Button Bar)
      Magic Hypercube Overview
      Hypercube Representations
      Hypercube Representations-2
      Order 3 Magic Tesseracts
      Hypercube Aspects
      Hypercube Classes
      Associated Hypercubes
      Hypercube Math
      More Tesseracts
      Hypercube Cross-stitch
      Knight Tours in Higher Dimensions
      The Unfolded Tesseract

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Top.gif (1256 bytes)Thanks for the visit and I hope to see you again soon.

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Harvey Heinz
This page last updated October 31, 2009
Copyright 1998, 1999, 2000, 2001, 2002 by Harvey D. Heinz