Magic Hypercubes Site Map



This page includes links to all of the pages on this large site. As mentioned elsewhere, this site is a consists of material originally published over a period of more then a decade and assembled online in 3 different sites. Very little effort has been made to reorganize the material from the original piece-by-piece assembly.

Hopefully, this page will be of assistance in understanding the layout and finding your way to similar material appearing in different areas. The numbers before the links indicate the order of the pages if using the back and forward arrows at the bottom of each page.

Squares Cubes Tesseracts Stars Numbers

Home Page for the entire site This is the introduction page for the entire site.
As well as these general pages, it has buttons linking to the 5 main section heads.
Bibliography More then 160 books, chapters of books or papers dealing with magic squares, etc.
Book for Sale A reference book on magic squares, cubes, tesseracts, magic stars, etc.
Downloads Some of my Word, Excel and Basic program files and some collected papers.
Glossary Approximately 125 terms related to magic squares, cubes, tesseracts, stars, etc.
Links 36 links to magic square sites and 25 links to other math sites.
Site Map This page
Search A search page for the entire site.

Magic Squares
     Home Page

An introduction to magic squares and some examples.
Links to the other pages in this section.

  1. A Deluxe Magic Square

An Order-5 Pandiagonal, Associative, Compact, and Self-similar Magic Square

  2. More Magic Squares

9 sections on very different magic squares circles and hexagons

  3. More Magic Squares-2

An assortment of 12 different types of magic squares

  4. Material From REC

Excerpts from Dr. M. Ecker's Recreational & Educational Computing Newsletter

  5. Unusual Magic Squares

12 very different magic squares

  6. John Hendricks - Cubes

14 magic hypercube constructions by the late John Hendricks

  7. Prime Number Magic Squares

22 sections about magic squares constructed with prime numbers

  8. Quadrant Magic Squares

A type of magic square that contains patterns in all 4 quadrants

      a. Order-13 Quadrant Magic Squares

Quadrant m.s. with various number of magic quadrants (11 sections)

      b. Order-17 Quadrant Magic Squares

10 sections as above

  9. Even Order Quadrant Magic Squares The even order equivalent of the above quadrant magic squares
      a. Order-16 Quadrant Magic Squares All 52 symmetrical patterns and 4 example order-16 QMS
10 . Type 2 Order-3 Magic Squares

An arrangement of non-consecutive numbers that can also form a magic square.

11 Anti-magic Squares

6 sections on a different type of magic

12  Self-similar Magic Squares

Squares that transform to another aspect of them self

13. Most-perfect Magic Squares

A special type of pandiagonal magic square (9 sections)

      a.  Most-perfect Bent-diagonals M. S.

A sub-set of the above where the bent-diagonals are also magic

14. Magic Square Models

Cross-stitch or wooden-block creations of magic squares, cubes, tesseracts, stars

15. Transformations and Patterns

9 sections on transforming one order-4 magic square to another one.

      a. More Order-4 Transformations

8 additional sections on this subject.

      b. Transformations summary

4 more sections including a table of 48 transformation procedures

      c. Fellows Transformations

4 additional methods supplied by Ralph Fellows

16. Order-4 Magic Squares

In-depth description of the 12 Dudeney groups of order-4 magic squares

      a. Order-4  # 1 to 200

List of the first 200 of the 880 order-4 magic squares in Frénícle index order

      b. Order-4  # 201 to 400

The next 200

      c. Order-4  # 401 to 600

The next 200

      d. Order-4  # 601 to 880

The last 280

17. Pandiagonal Order-5 Magic Squares

The 36 essentially different order-5 pandiagonal magic squares to the 3600 total

18. Franklin Squares

Franklin squares - not all magic (13 sections)

19. Multimagic Squares

Regular magic squares that remain magic when each number is raised to a power

20. Perimeter Magic Triangles

6 sections on these different magic figures

21. Perimeter Magic Polygon >k=3

Perimeter magic squares, pentagons, hexagons, and a method of construction

22. Magic 3-D Polygons and Graphs

14 sections on more elaborate magic figures

23. Perimeter Magic Platonic Solids Bao Qi-shou circa 1880 diagrams
24. Magic Knight Tours

These 'magic' tours may result in only semi-magic squares

25. Compact Magic Squares

Magic squares where all 2x2 sub-squares sum to 4/m of S

26. Ultra-magic Squares

Material from Walter Trump where all squares contain unusual patterns

27. Square-Update A late 2009 update. Luo Shu format, Topological, and Totally Irregular squares,
28. Square-Update-2 Posted July 2010. A variety of new information and old unique items.
29. Sparse Magic Squares Some samples of some magic squares with several cells empty.

Magic Cubes
     Home Page

A brief introduction to this section and several other sections.
Then a Table of Contents with brief descriptions and links to the following pages.
  1. Magic Cubes – The Basics Magic cube parts, associated, basic cube and aspects, coordinates, species.
  2. Magic Cube Definitions A discussion of common terms relating to magic cubes.
  3. Perfect Magic Hypercubes My original page on perfect (nasik) magic cubes.
  4. Magic Cubes – The Road to Perfect The progression to perfect (nasik) cubes. Presented is 1 cube each of orders 3 to 11.
  5. The Early Cubes 15 different cubes from Fermat's 1640 order 4 to Worthington's 1910 order 6.
  6. A. H. Frost’s Magic Cubes 8 assorted  cubes published by Rev. Frost in 1866 and 1878.
  7. F. A. P. Barnard’s Magic Cubes His perfect (nasik) orders 8 and 11. And other magic objects from his 1888 paper.
  8. Magic Cubes – Order 3 The only four order 3 basic cubes, and some variations. Other material.
  9. Magic Cubes – Order 4 A number of order-4 cubes with differing characteristics.
10. Magic Cubes – Order 5 8 different cubes, (1876 to 2001).The Trump/Boyer diagonal order 5 cube of 2003.
11. Magic Cubes – Order 6 A variety of 7 cubes, published between 1838 and 1999.
12. Magic Cubes – Order 7 A variety of 7 cubes, published between 1922 and 2001.
13. Magic Cubes – Order 8 A variety of 6 cubes, published between 1908 and 2001.
14. Magic Cubes – Order 9 Three simple magic cubes, all with slightly different features.
15. Magic Cubes – Order 10 Three simple magic cubes, one of them with an order 6 inlaid cube.
16. Magic Cubes – Order 11 A simple, a pantriagonal, and 2 perfect order 11 cubes.
17. Magic Cubes – Order 12 A pantriagonal, a diagonal, and a simple, but inlaid, order 12 magic cube.
18. Magic Cubes – Order 13 A pantriagonal and a perfect cube. Also, an example of a broken plane.
19. Arnoux’s Perfect Cube 2 different order-15 and 2 different order-16. Arnoux's order-17 of  1887!
20. Arnoux Patterns Patterns that it turns out are present in a wide variety of magic cubes.
21. Modulo Magic Cubes Seven order 5 cubes with line sums evenly divisible by the same number.
22. Multimagic Cubes Presenting the world's first Bimagic and Trimagic cubes
     a. Boyer's Monster Cubes Christian Boyer announcing further advances in multimagic cubes and tesseracts.
     b. Boyer’s Bimagic Order 16 Cube The complete listing  Boyer's bimagic order 16 cube of Jan. 23, 2003
     c. Boyer’s Bimagic Order 32 Cube The top horizontal plane of  Boyer's bimagic order 32 cube of Jan. 27, 2003
23. Magic Cubes - Groups Dudeney groups I to VI magic squares and their magic cube equivalents.
24. Prime Number Magic Cubes Two order 3 prime cubes. An order 4 simple cube, and an order 4 pantriagonal.
25. Magic Cubes – Multiply Three different types of order 3 multiply cubes. An order 4 and an order 5 cube. etc.
26. Composition Magic Cubes An order-9 and an order 12 cubes made up of smaller cubes. etc.
27. Hendricks` Inlaid Magic Cubes Cubes (and a tesseract) with multiple inlaid cubes and squares. 5 sections + 1 link.
28. The Heinz X6 Magic Cube Description, pictures, and listings of my wood block model of 6 order 4 cubes in one.
29. Self-similar Magic Cubes Different types of symmetrical cubes. Thanks Walter Trump!
30. Magic Cubes – Pan and Semi-pan A description & comparison of pandiagonal and semi-pandiagonal squares & cubes.
31. Unusual Magic Cubes About 15 cubes that are not magic in the ordinary sense, but are unusual!
32. Most-Perfect Magic Cubes Discussion and examples of the 3-D equivalent of the most-perfect magic square.
     a. Order-16 Perfect Cubes Listings of two order 16 perfect magic cubes. Only one is most-perfect.
33. Summary of this Magic Cube Site Concluding remarks, amazing new advances in magic cube knowledge, & challenges!
34. Magic Cube Update-1 Material that I received in January, 2004. (Heterocube, Purely Pan cube, etc.
35. Magic Cube Update-2 Material received to April 30, 2004. (Cubes (1757), Order 6 Projection cube, etc.
36. Magic Cube Update-3 Received during the last nine months of 2004. Nested order 16, A new class, etc.
37. Magic Cube Update-4 Aug. 2005. More on Panmagic ratios, Semi-diagonal magic order 4, etc.
38. Magic Cube Update-5 May  2007. Magic cuboids, Magic Knight Tours, transforms, links, etc.
39. Magic Cube Update-6 Feb. 2010. Frost Order-9 model. More on Compact & Complete. Multiply order-4.
40. Cube Timeline 5 sections, including First cube of each class for each order, Timeline, References

Magic Tesseracts
            Home Page

This page has a brief description of what a tesseract is, and illustrates the main parts.
Then descriptions and links to the remaining pages in this section.
1.  Magic Hypercubes - Overview A general overview of magic squares, cubes, tesseract, etc. Their interrelationships and summations
2.  Hypercube Representations A review of how magic squares and cubes have been presented in publications, etc. through the ages.
3.  Hypercube Representations - 2 A review of how the forth dimension and magic tesseracts have been illustrated (over a shorter time period).
4.  Order-3 Magic Tesseracts Features of order 3 hypercubes and illustrations and listing for all 58 basic order-3 tesseracts
5.  Hypercube Aspects Illustrates the 8 aspects of the magic square and 48 aspects of the magic cube. The tesseract has 384 aspects
6.  Hypercube Classes Reviews classes of magic squares and cubes. Minimum requirements for the 18 of magic tesseract classes
7.  Associated Hypercubes Discusses features of  associated magic hypercubes, and embedded hypercubes of lower dimensions
8.  Hypercube Math Equations and comparison tables. Demonstrates close relationship between hypercubes of different dimensions
9.  More Tesseracts Orders 5 and 6 tesseracts, Inlaid tesseract, perfect tesseracts, plus links to some previously posted on other sites
10. Hypercube - Cross-stitch A picture and discussion of a cross-stitch project demonstrating features of square, cube, and tesseract
11. Tesseract Knight Tour Awani Kumar has successfully constructed an order 4 tesseract containing a magic knight tour
12. The Unfolded Tesseract Showing the 3-D illustration of a 4_D tesseract flattened out to 2 Dimensions to show all 24 faces
13. Tesseracts - Update 2013 Hypercube Summary Table - Recent developments - Contributors to Magic Tesseract theory  - The 5-D Challenge

Magic Stars
    Home Page

Contains a basic definition of Magic Stars and the similarity to Magic Squares.
Also links to other pages in this section.   Off the main topic, but one of two original postings!
  1. Magic Stat Definitions Defines magic stars in more detail divided into 9 sections
  2. Magic Star Examples 16 example diagrams of magic stars of orders 5 to 11
  3. Magic Star Examples-2 Examples of the 14 patterns of orders 12 to 14
  4. Big Magic Stars 7 examples of some patterns of m. stars of orders 15 to 20 (Simon Whitechapel)
  5. Order-5 Magic Stars 9 sections on order-5 m. stars, including almost-magic, anti-magic, tree-planting
  6. Order-6 Magic Stars Characteristics, early solutions solvers, twenty sets of four, super-magic stars, etc.
     7. Order-6 Solution List A tabular listing of the 80 basic solutions along with additional features
     8. H. E. Dudeney Features Features of the order-6 m. stars that were first reported by H. E. Dudeney in 1926
  9. Order-7 Magic Stars Features and a tabular listing of the 72 basic solutions of each of the 2 patterns
10. Order-8 Magic Stars Features and a tabular listing of the 112 basic solutions of each of the 2 patterns
11. Order-9 Magic Stars Features and partial lists of 3014 pattern A and 1676 patterns B and C solutions
12. Order-10 Magic Stars Features and partial lists of 10,882 patterns A and C, and 115,552 pattern B
13. Order-11 Magic Stars Features and partial lists of  of patterns A, B, C, and D.
14. Prime Number Magic Stars All basic solutions for orders 5 and 6 minimal and consecutive primes
15. Prime Number Magic Stars-2 All basic solutions for orders 7 and 8 minimal and consecutive primes
16. Unusual Magic Stars 12 sections, such as five-in-one, multiplying hexagram, 6 in 6, 8 in 8, etc.
17. Isomorphic Magic Stars 8 sections about magic stars derived from magic squares
18. Magic 9 x 5 Hexagrams Special 6 pointed stars composed of 9 lines of 5 numbers
19. 3 - D Magic Star The only possible (?)  3 dimensional magic star has 12 lines of 3 numbers
20. Trenkler Stars A special class of magic star (type T). Also 'weakly' and almost-magic'
21. Magic Star Puzzles Photographs and diagrams of magic star puzzles published over time
22. Star Updates Tree planting problems and a fractional star
      a.   Simon Whitechapel Some of his investigations into magic stars (mostly big stars)
      b.  Jon Wharf Jon confirmed my counts for the orders 5 - 11 using different methods
      c.  Andrew Howroyd Andrew also confirmed counts for 6-13. He investigated orders 10-11 permutations

Magic Numbers
         Home Page
An introduction and some general number patterns.
Links to other pages in this section. Off the main topic, but one of two original postings!
1. Prime Patterns 45 patterns consisting of prime numbers
2. More Prime Patterns 21 sections. Most have several related patterns. One is of prime number plots
3. Narcissistic Numbers 24 sections. Most have several related patterns.
4. Palindromes 7 sections devoted to palindromic numbers.


This page was originally posted _____
It was last updated March 23, 2013
Harvey Heinz
Copyright © 1998-2009 by Harvey D. Heinz