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Contents
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This magic square
contains an order-3 and an order-4 magic diamond. |
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Orders 3and 4 interleaved
magic squares and other properties. |
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Inlaid are an order-7 and
an order-5 and also an order-3 diamond. |
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Included here are orders
3 and 5 magic squares and magic diamonds. |
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A total of 15 magic
squares in one. Orders 4, 7, 8, and 15. |
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One has 10 magic squares
with bent diagonals & 1 has 5 pandiagonals. |
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Composed of nine order-5
magic squares, each with a magic diamond. |
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A normal order-3 magic
cube using numbers 1 - 27. |
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This order-4 magic cube
has all broken pan-triagonals summing correctly. |
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This magic hypercube is
4-dimensional and uses numbers 1 to 81. |
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Contains an order-4
pan-triagonal cube and 12 pandiagonal magic squares |
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Announcement of world's
first inlaid magic tesseract. Oct. 15/99. |
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A bimagic square by David
Collison and a new type by John Hendricks. |
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A brief autobiography and
outline of accomplishments in this field. |
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John Hendricks original
web page (now maintained by H. Heinz) |
Order-7 with Diamond Inlays
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Order-7 Magic Square S = 175 Order-4 Magic
Diamond S = 100
Order-3 Magic Diamond S = 75
From The Magic Square Course, title page for chap.
XII
(See bibliography at end of this page.)
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Order-7 with Square Inlays
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Order-7 Magic Square sum = 175
Order-4 (pink) Magic Square sum = 100
Order-3 (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 |
Order-9 with Diamond Inlays
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S9 = 369
S7 = 287
S5 = 205
S3 = 123
This inlay may be used in place of the above order-5
inlay
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42 |
34 |
49 |
30 |
50 |
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22 |
39 |
61 |
23 |
60 |
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51 |
25 |
41 |
57 |
31 |
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58 |
59 |
21 |
43 |
24 |
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32 |
48 |
33 |
52 |
40 |
From The Magic Square
Course, page 191 |
Order-14 Ornamental
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For a total of 9 magic squares.
By John Hendricks (unpublished)
| MAGIC SUMS
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S14 = 1379
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| Top |
Sub-square
S5 = 615
S3 = 369 |
Diamond
S4 = 100
S3 = 516 |
| Bottom
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Sub-square
S5 = 370
S3 = 222 |
Diamond
S4 = 688
S3 = 75 |
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Order-15 Overlapping
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For a total of fifteen Magic squares
From The Magic Square Course,
page 232
MAGIC SUMS S15 =
1695
Lower left & upper right
2 x S7 = 791 (pandiagonal)
Upper left & lower right
2 x S8 = 904
10 x S4 = 452
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Two related ten-in one
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This order-8 magic square contains four order-4
magic squares in the quadrants and one order-4 in the center.
It contains four more order-4 magic squares starting with the top
left hand corner at 24, 59, 27 and 20 (outlined in blue).S4
= 130 S8 = 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 order-4 are 1 + 64 + 3 + 62 and
1 + 64 + 17 + 48.
An order-8 bent diagonal is 44 + 21 + 7 + 58 + 37 + 28 + 10 + 55.
Figure 12a from Inlaid
Magic Squares & Cubes, page 18 |
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This order-8 magic square is pandiagonal.
The four order-4 magic squares in the quadrants are also
pandiagonal.
The order-4 in the center and the four order-4 outlined in blue
are regular magic squares.
Figure 11h from Inlaid
Magic Squares & Cubes (revised), page 17 |
Order-15 Composite
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This order-15 magic square consists of 9 order-5
magic squares, each with an order-3 inlaid diamond magic square.
As is common with composite magic squares, the magic sums of the
order-5 squares themselves form an order-3 magic square with the
constant 1695.
The constants of the order-3 magic diamonds form an order-3 magic
square with the constant 1017.
S of Order-5 squares
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560 |
645 |
490 |
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495 |
565 |
635 |
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640 |
485 |
570 |
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S of Order-diamonds
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336 |
387 |
294 |
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297 |
339 |
381 |
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384 |
291 |
342 |
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From The Magic Square
Course, page 244 |
Order-3 Magic Cube
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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, order-8 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 3-agonals, 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 order-3 magic square is associated, so also are
the four order-3 magic cubes. Because they are associated, all are
also self-similar. That is, when each number is subtracted from 28
the result is a reflection of itself. See my
Self-similar Magic Squares.
From The Magic Square
Course, page 329. |
Pan-3-agonal magic
cube
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16 rows sum to 130
16 columns sum to 130
16 pillars sum to 130
Four 3-dimensional diagonals sum to 130.
All broken 3-agonals 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 pan-3-agonal magic cube.
The numbers circled in red show one of the 4 main triagonals.
In an order-4 cube it is impossible for all the diagonals parallel
to the faces to be magic.
John Hendricks coined the term pan-3-agonal for the broken space
3- agonals.
There are 7680 pan-3-agonal magic cubes of order-4. The total number
of Order-4 magic cubes is not known. |
From The Magic Square Course,
page 384.
This also appeared in The Journal of Recreational Mathematics, 5(1) p.
51-52.
Order-3 Magic Tesseract
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This order-3 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 4-agonals, 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 order-3 magic square and the four order-3 magic cubes
are associated, so also are the 58 order-3 magic tesseracts. Because they
are associated, all are also self-similar. 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
Self-similar Magic Squares.
From The Magic Square Course,
page 470-491 (which shows all 58 order-3).
Order-8 Inlaid Magic Cube
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Here is the shell for an order-8 magic cube with an
inlaid order-4 magic cube.
Each of the six windows shown holds two order-4 pandiagonal magic
squares.
The inner order-4 cube is pantriagonal meaning that all broken
triagonal pairs sum correctly to 1026.
The order-8 cube uses the numbers from 1 to 83 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 order-8 (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 semi-magic (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
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# 308
- 151
St. Andrews St.,
Victoria, B.C., V8V 2M9
CANADA
15 October, 1999
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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
tenth-order magic square which sums 505 in rows columns and diagonals. In
the second quadrant, you will find inlaid a 5th-order 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 four-dimensional 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. |
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John R. Hendricks |
Order-9 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 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.
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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 in-service 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 six-dimensional 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 Order-9, Dec. 1999, 14 pages 8 ½ x
11+covers 0-9684700-6-8
John R. Hendricks, Perfect n-Dimensional Hypercubes of Order 2n,
May 1999, 36 pages 8 ½ x 11, 0-9684700-4-1
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, 0-9684700-1-7
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, 0-9684700-2-5
John R. Hendricks, Magic Squares to Tesseracts
by Computer, 1998, 212 pages 8 ½ x 11, 0-9684700-0-9
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 e-mail address.Sadly, John Hendricks passed away on
July 7, 2007. I am still maintaining (Sept. 1, 2009) his original web site
here. |
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