Order7 with Diamond Inlays 
This magic square contains an order3 and an order4 magic diamond. 
Order7 with Square Inlays 
Orders 3and 4 interleaved magic squares and other properties. 
Order9 with Diamond Inlays 
Inlaid are an order7 and an order5 and also an order3 diamond. 
Order14 Ornamental 
Included here are orders 3 and 5 magic squares and magic diamonds. 
Order15 Overlapping 
A total of 15 magic squares in one. Orders 4, 7, 8, and 15. 
Two related tenin one 
One has 10 magic squares with bent diagonals & 1 has 5 pandiagonals. 
Order15 Composite 
Composed of nine order5 magic squares, each with a magic diamond. 
Order3 Magic Cube 
A normal order3 magic cube using numbers 1  27. 
Pan3agonal magic cube 
This order4 magic cube has all broken pantriagonals summing correctly. 
Order3 Magic Tesseract 
This magic hypercube is 4dimensional and uses numbers 1 to 81. 
Order8 Inlaid Magic Cube 
Contains an order4 pantriagonal cube and 12 pandiagonal magic squares 
Inlaid Magic Tesseract 
Announcement of world's first inlaid magic tesseract. Oct. 15/99. 
Order9 Bimagic Square 
A bimagic square by David Collison and a new type by John Hendricks. 
About John Hendricks 
A brief autobiography and outline of accomplishments in this field. 
JohnHendricksMath 
John Hendricks original web page (now maintained by H. Heinz) 
Order7 Magic Square S = 175
Order4 Magic Diamond S = 100
Order3 Magic Diamond S = 75
From The Magic Square Course, title page for chap. XII
Order7 Magic Square
sum = 175
Order4 (pink) Magic Square sum = 100
Order3 (blue) Magic Square sum = 75
Corners of each of these 3 magic squares sum to 100
Uncolored squares sum as follows (in any direction):
lines of 2 cells = 50
lines of 3 cells = 75
lines of 4 cells = 100
lines of 6 cells = 150
From The Magic Square Course, page 46
S_{9} = 369
S_{7} = 287
S_{5} = 205
S_{3} = 123
This inlay may be used in place of the above order5 inlay
42 
34 
49 
30 
50 
22 
39 
61 
23 
60 
51 
25 
41 
57 
31 
58 
59 
21 
43 
24 
32 
48 
33 
52 
40 
From The Magic Square Course, page 191
MAGIC SUMS S_{14} = 1379
TOP Subsquare
Diamond
Bottom Subsquare
Diamond
S_{5 }=
615_{
}S_{4} = 100
S_{5} =
370 S_{4}
= 688
S_{3} = 369
S_{3} =
516
S_{3} =
222 S_{3}
= 75
For a total of 9 magic squares. By John Hendricks (unpublished)
MAGIC SUMS Lower left
& upper right Upper
left & lower right
S_{15} = 1695
2 x S_{7} = 791 (Pandiagonal)
2 x S_{8} = 904
10 x S_{4} = 452
For a total of fifteen Magic squares
From The Magic Square Course, page 232
This order8 magic square contains four order4 magic squares in the quadrants and one
order4 in the center. It contains four more order4 magic squares starting with the top left hand corner at 24, 59, 27 and 20 (outlined in blue). S_{4} = 130 S_{8} = 260 All ten of these magic squares have the additional feature that each of the four bent
diagonals also sum correctly. Two of these bent diagonals in the top left hand order4 are
1 + 64 + 3 + 62 and 1 + 64 + 17 + 48. Figure 12a from Inlaid Magic Squares & Cubes, page 18 

This order8 magic square is pandiagonal. The four order4 magic squares in the
quadrants are also pandiagonal. The order4 in the center and the four order4 outlined in blue are regular magic squares.
Figure 11h from Inlaid Magic Squares & Cubes (revised), page 17 
S of Order5 squares

S of Order3 diamonds

This order15 magic square consists of 9 order5 magic
squares, each with an order3 inlaid diamond magic square. As is common with composite
magic squares, the magic sums of the order5 squares themselves form an order3 magic
square with the constant 1695. From The Magic Square Course, page 244 
9 rows, such as 1  17  24, sum to 42.
9 columns, such as 1  15  26, sum to 42.
9 Pillars, such as 1  23  18, sum to 42.
4 triagonals, such as 26  14  2 sum to 42.
Some of the squares may have diagonals summing to 42, but this is not a requirement. In fact, order8 is the smallest cube for which it is possible for all the diagonals to sum correctly.
What is required is that the 4 triagonals or 3agonals, such as 1  14  27 sum to 42.
There are 4 different basic pure (using numbers 1 to 27) magic cubes. Each of these have 48 equivalents due to rotations and/or reflections.
From The Magic Square Course, page 329.
Just as the one order3 magic square is associated, so also are the four order3 magic
cubes. Because they are associated, all are also selfsimilar. That is, when each number
is subtracted from 28 the result is a reflection of itself.
See my Selfsimilar Magic Squares.
16 rows sum to 130
16 columns sum to 130
16 pillars sum to 130
Four 3dimensional diagonals sum to 130.
All broken 3agonals parallel to the 4 main triagonals also sum to 130.
This is the equivalent to the pandiagonal Magic Square.
Because it is pandiagonal, any face may be moved to the opposite side, thus
creating a new pan3agonal magic cube.
The numbers circled in red show one of the 4 main triagonals.
In an order4 cube it is impossible for all the diagonals parallel to the faces to be
magic.
John Hendricks coined the term pan3agonal for the broken space 3 agonals.
There are 7680 pan3agonal magic cubes of order4. The total number of Order4 magic
cubes is not known.
From The Magic Square Course, page 384
This also appeared in The Journal of Recreational Mathematics, 5(1) p. 5152.
This order3 hypercube of four dimensions is shown in two dimensions using lines to depict the outer dimensions only. The colored numbers here show the middle cube in the horizontal plane. There are three cubes also in each of the other three planes.
27 rows, such as 50  12  61, sum to 123.
27 columns, such as 50  72  1, sum to 123.
27 pillars, such as 50  64  9, sum to 123.
27 files, such as 50  16  57, sum to 123.
8 quadragonals, such as 1  41  81 sum to 123.
Some of the squares may have diagonals summing to 123 and some of the cubes may have
triagonals summing to 123. These are not requirements of a magic tesseract just as a magic
cube is not required to have the planar square diagonals summing to 42.
What is required is that the 8 quadragonals or 4agonals, such as 50  41  32 sum to 123.
There are 58 different basic pure (using numbers 1 to 81) magic tesseracts. Each of
these have 384 equivalents due to rotations and/or reflections.
From The Magic Square Course, page 470491 (which shows all 58 order3).
Just as the one order3 magic square and the four order3 magic cubes are associated, so also are the 58 order3 magic tesseracts. Because they are associated, all are also selfsimilar. That is, when each number is subtracted from 82 (i.e. complimented), the result is a reflection of itself and is one of the 384 aspects of this figure. See my Selfsimilar Magic Squares.
Here is the shell for an order8 magic cube with an inlaid
order4 magic cube.
Each of the six windows shown holds two order4 pandiagonal magic squares.
The inner order4 cube is pantriagonal meaning that all broken triagonal pairs sum correctly to 1026.
The order8 cube uses the numbers from 1 to 8^{3} and has the magic sum of 2052.
Note that it is not a requirement that planar diagonals sum correctly for a cube to be considered magic, although it is possible for an order8 (the smallest order cube) to have this feature .
The author reasons that there are 2,717,908,992 variations of this one cube, obtainable by rotations, reflections and transformations of the components.
Here are the individual layers of the cube. Note that in most cases they are only semimagic (the planar diagonals will not sum correctly).
From the above eight horizontal planes, the 16 vertical planes and the four triagonals can be assembled.
From The Magic Square Course, pp. 419  431


Discovered during the spring of 1999, was a new method of making magic squares of order 2k. An example shown top left is a tenthorder magic square which sums 505 in rows columns and diagonals. In the second quadrant, you will find inlaid a 5thorder magic square which sums 315. Inlaid squares and various methods abound, so this simply adds another method into the system. MAJOR ANNOUNCEMENT The technique mentioned above, can be extended to three and fourdimensional space and higher. A magic tesseract of order six, with an Inlaid magic tesseract of order three has been made. It contains the numbers from 1 to 1296 and sums 3,891 in the required 872 different ways. This is the world’s first magic tesseract of order six. The inlaid magic tesseract of order three sums 1,824, in the required 116 different ways. This becomes the world’s first inlaid magic tesseract. The new method for magic squares will be taken into account in the upcoming Second Edition of Inlaid Magic Squares and Cubes, which is unscheduled at the moment. 

John R. Hendricks 
HENDRICKS’ BIMAGIC SQUARE OF ORDER NINE
Bimagic means that you can sum the numbers as they are, or you can square them all first and then sum them. Either way, the square is magic.
David M. Collison, first discovered bimagic squares of order nine. An example is shown in Figure 1. He died before he could reveal just how he made it and mathematicians are still searching to find his method of construction.
Figure 1. Collison’s regular, Figure
2. Hendricks’ newly
or associated bimagic square. created
bimagic variety.
In Figure 2, the square is partitioned into nine zones. These are not magic sub—squares — just zones. Each zone of nine elements sums 369, as does’ each row, each column and both diagonals. If you square all the numbers and then add them up, you will find that each zone sums 20,049, as does each’ row, column and both diagonals.
With the new bimagic square, you can translocate any 3 by 9 rectangle of numbers to the opposite side of the square, as shown above, and a new bimagic square will emerge.
John R. Hendricks
Victoria, BC, Canada
25 November 1999
Mr. Hendricks worked for the Canadian Meteorological Service for 33 years, and retired in 1984. Early in his career, he was a meteorological instructor for the N.A.T.O. Training Program. Later, he was a weather forecaster at various locations across Canada. Throughout his career, he was also known for his contributions to statistics and climatological statistics.
While employed, he also participated in volunteer service groups, including The Monarchist League of Canada and he was the founding President, Manitoba Provincial Council, The Duke of Edinburgh's Award in Canada. He was a recipient of the Canada 125 medal for his volunteer work.
Following his career in meteorology, he gave many public lectures on magic squares and cubes in schools and at inservice teacher's conventions both in Canada and in the northern United States. He developed a course on magic squares and cubes for the mathematically inclined students at Acadia Junior High School in Winnipeg for seven years. The resulting text book of over 550 8.5" x 11" pages was never published. He delivered half a dozen colloquia to professors of mathematics on the subject and in geometry and statistics, as well. He assisted in the Shad Valley program for several years.
John Hendricks started collecting magic squares and cubes when he was 13 years old. This became a hobby with him and sometimes even an obsession. He never really thought that he would ever expand the knowledge in this field. But soon, he became the first person in the world to successfully make four, five and sixdimensional models of magic hypercubes, and publish them. He has written prolifically on the subject in the Journal of Recreational Mathematics. He has also extended the knowledge of magic squares and cubes, especially the ornate and embedded varieties.
John R. Hendricks, Bimagic Squares of Order9, Dec.
1999, 14 pages 8 ½ x 11+covers 0968470068
John R. Hendricks, Perfect nDimensional Hypercubes of Order 2^{n}, May
1999, 36 pages 8 ½ x 11, 0968470041
Equations are shown for the first
perfect Tesseract and Basic programs for orders 4 6.
John R. Hendricks, Inlaid Magic Squares and Cubes, Feb.
1999, 214 pages 8 ½ x 11, 0968470017
Equations, examples, programs and a list of
the 46 articles (mostly magic square related) he has had published in journals.
John R. Hendricks, All Third Order Magic Tesseracts, Feb. 1999,
36 pages 8 ½ x 11, 0968470025
John R. Hendricks, Magic Squares to Tesseracts by
Computer, 1998, 212 pages 8 ½ x 11, 0968470009
Equations, examples, and 3 appendices dealing
with rotations/reflections, magic squares of order 4k+2, and programs.
Update: January 30, 2004 Due to ill health and depleted stock, John's books are no longer available. He no longer has an email address. Sadly, John Hendricks passed away on July 7, 2007. I am still maintaining (Sept. 1, 2009) his original web site here. 
Please send me Feedback about my Web site!
Last updated
September 01, 2009
Harvey Heinz harveyheinz@shaw.ca
Copyright © 1998, 1999,2000 by Harvey D. Heinz