# 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.)

Moved to
magic-squares.net
A few pages of this collection were first posted to Geocities in March 1998 as www.geocities.com/CapeCanaveral/Launchpad/4057/. Over the years the collection grew in size and became www.geocities.com/~harveyh/. 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 http://recmath.org/

### 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.

## 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.

Some of my Word, Excel and Basic program files.

### 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.

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.

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 http://members.shaw.ca/hdhcubes/, and recently the tesseract site mentioned above.

### 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

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 http://uk.groups.yahoo.com/group/magicsquares/

Magic Cubes Newsgroup at http://groups.yahoo.com/group/magiccubes/
For magic Squares of Luo Shu format, see Dr. Robert Dickter's http://www.luo-shu.com/
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 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/
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  http://www.aaamath.com/
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-É. 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  http://www.numberspiral.com/
Math for Kids at http://www.allmath.com is an entry point to many web sites with interesting mathematical content.
David Singmaster's source lists on Recreational Mathematics at www.g4g4.com/MyCD5/SOURCES/singmaterial.htm

### 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
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
Simon Whitechapel
Jon Wharf
Andrew Howroyd
Number Patterns
Prime Patterns
More Prime Patterns
Narcissistic Numbers
Palindromes
Glossary
Bibliography
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
Summary of this Magic Cube Site
Cube Update-1
Cube Update-2
Cube Update-3
Cube Update-4
Cube_Update-5
Cube_Timeline```
```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```

Thanks for the visit and I hope to see you again soon.