|
  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 |
