Contents

This magic square
contains an order3 and an order4 magic diamond. 

Orders 3and 4 interleaved
magic squares and other properties. 

Inlaid are an order7 and
an order5 and also an order3 diamond. 

Included here are orders
3 and 5 magic squares and magic diamonds. 

A total of 15 magic
squares in one. Orders 4, 7, 8, and 15. 

One has 10 magic squares
with bent diagonals & 1 has 5 pandiagonals. 

Composed of nine order5
magic squares, each with a magic diamond. 

A normal order3 magic
cube using numbers 1  27. 

This order4 magic cube
has all broken pantriagonals summing correctly. 

This magic hypercube is
4dimensional and uses numbers 1 to 81. 

Contains an order4
pantriagonal cube and 12 pandiagonal magic squares 

Announcement of world's
first inlaid magic tesseract. Oct. 15/99. 

A bimagic square by David
Collison and a new type by John Hendricks. 

A brief autobiography and
outline of accomplishments in this field. 

John Hendricks original
web page (now maintained by H. Heinz) 
Order7 with Diamond Inlays

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
(See bibliography at end of this page.)

Order7 with Square Inlays

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 
Order9 with Diamond Inlays

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 
Order14 Ornamental

For a total of 9 magic squares.
By John Hendricks (unpublished)
MAGIC SUMS

S_{14} = 1379

Top 
Subsquare
S_{5 }= 615
S_{3} = 369 
Diamond
S_{4} = 100
S_{3} = 516 
Bottom

Subsquare
S_{5} = 370
S_{3} = 222 
Diamond
S_{4} = 688
S_{3} = 75 

Order15 Overlapping

For a total of fifteen Magic squares
From The Magic Square Course,
page 232
MAGIC SUMS S_{15} =
1695
Lower left & upper right
2 x S_{7} = 791 (pandiagonal)
Upper left & lower right
2 x S_{8} = 904
10 x S_{4} = 452

Two related tenin one

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.
An order8 bent diagonal is 44 + 21 + 7 + 58 + 37 + 28 + 10 + 55.
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 
Order15 Composite

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.
The constants of the order3 magic diamonds form an order3 magic
square with the constant 1017.
S of Order5 squares
560 
645 
490 
495 
565 
635 
640 
485 
570 

S of Orderdiamonds
336 
387 
294 
297 
339 
381 
384 
291 
342 

From The Magic Square
Course, page 244 
Order3 Magic Cube

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.
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.
From The Magic Square
Course, page 329. 
Pan3agonal magic
cube

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.
Order3 Magic Tesseract

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. 
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.
From The Magic Square Course,
page 470491 (which shows all 58 order3).
Order8 Inlaid Magic Cube

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.
Following are listed the individual layers of the cube. Note
that in most cases they are only semimagic (the planar diagonals
are not required to sum correctly for the cube to be considered
magic). 
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
Inlaid Magic Tesseract

# 308
 151
St. Andrews St.,
Victoria, B.C., V8V 2M9
CANADA
15 October, 1999

Minor Announcement
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 
Order9 Bimagic Square
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 subsquares, 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 at left, and a new bimagic square will emerge. 
John R. Hendricks
Victoria, BC, Canada
25 November 1999
About John Hendricks
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, The Magic Square Course,
Unpublished, 1991, 554 pages 8.5 “ x 11”.
Written for a high school math
enrichment class he conducted for 5 years.
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. 
