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IN MEMORIAM:
JOHN ROBERT HENDRICKS
September 4, 1929 – July 7, 2007
John Hendricks was born in Regina, Saskatchewan, Canada, but grew up in
Vancouver, British Columbia. He saw his first magic square at the age of
13 and this proved to be the start of a lifelong hobby. He graduated from
the University of British Columbia in 1951 with a degree in mathematics.
He worked for the Canadian Meteorological Service until his retirement in
1984, in Winnipeg, Manitoba.
John participated in many volunteer groups. He was the founding President,
Manitoba Provincial Council, The Duke of Edinburgh’s Award in Canada, and
assisted with the Shad Valley program. He received the Canada 125 medal,
in recognition of significant contribution to community and to Canada,
from the Lieutenant Governor of Manitoba on October the 19th, 1993.
John is known for his many published articles in meteorology, statistics
and statistical climatology (at least 29). But the greatest preponderance
of work was devoted to the study of magic squares, cubes and hypercubes.
Since retirement, he developed a magic square course for Junior High
students; spoke at teachers’ in-service sessions, spoke at university
colloquia, etc.
In 1962, John published a paper The Five and Six-Dimensional Magic
Hypercubes of Order 3 in The Canadian Mathematical Bulletin in which
he introduced a much needed method of illustrating the 4-dimensional
hypercube (the tesseract). Subsequently, he published over 45 articles on
magic squares, cubes, etc in The Journal of Recreational Mathematics, as
well as a small number in other math and educational journals. He also
self-published a number of books on the subject. Much of his material is
now in the University of Calgary Strens Recreational Mathematics
collection.
He was the first to construct and project onto a piece of paper the four,
five, and six dimensional magic hypercubes of order three. He was the
first to exhibit all 58 magic tesseracts of order 3. He proved there were
only 4 basic magic cubes of order 3 and supplied several methods of
construction of odd-ordered magic cubes, including a diagonal rule for
cubes. He developed methods for creating inlaid magic squares, cubes, and
tesseracts. All of this was accomplished with the aid of a hand held
computer only. (He obtained his first desk top computer in about 1999, and
used it mostly for e-mail and web surfing.
Since 1999 John has constructed
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the first order 16 perfect magic tesseract (Apr. 1999) |
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the first order-32 perfect magic tesseract (1999) |
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the first inlaid magic tesseract (order-6 with inlaid order-3,
Oct.,1999); and |
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the first bimagic cube (order-25, June 2000) |
In his later years John suffered from Parkinson’s Disease. By 2004 he was
virtually incapacitated with it and was hospitalized and near death
several times. He passed away in Victoria, BC, Canada on July 7, 2007.
Journal of Recreational Mathematics, Vol. 34:1, 80-81,
2005-2006. , © 2007 Baywood Publishing Co., Inc.
Reproduced by permission (July 7, 2008)
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Although efforts
were made by many people to extend magic squares and cubes into 4-dimensional
space prior to World War 2, very little happened until
John Hendricks discovered 5- and 6-Dimensional Magic
Hypercubes of order 3, in a paper published in 1962.
His work is now
spread across 4 or 5 different websites, as well as shown here. The websites are
linked with minimum duplication.
Contents
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Area of a polygon,
circumference of an ellipse, binodal lemniscate, etc. |
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A discussion concerning
diagonal intersections in hypercubes. |
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Substance of a talk given to
the Annual General Meeting of the Statistical Association of Manitoba |
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A history and explanation. Then
pages on Inlaid and Perfect Tesseracts |
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A description of recent books
published by J. R. H. , an order form and a bibliography. |
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List all the pages, with
relationships and links. |
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An order 25 bimagic cube
presented June 9, 2000 |
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Posters and Documents suitable
for high school math use and available for downloading. |

 John R.
Hendricks |
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Count the number of magic squares |
|
by John R. Hendricks |
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94 |
57 |
76 |
63 |
142 |
9 |
124 |
15 |
118 |
33 |
100 |
39 |
|
75 |
64 |
93 |
58 |
123 |
16 |
141 |
10 |
99 |
40 |
117 |
34 |
|
69 |
82 |
51 |
88 |
21 |
130 |
3 |
136 |
45 |
106 |
27 |
112 |
|
52 |
87 |
70 |
81 |
4 |
135 |
22 |
129 |
28 |
111 |
46 |
105 |
|
96 |
59 |
74 |
61 |
144 |
11 |
122 |
13 |
120 |
35 |
98 |
37 |
|
73 |
62 |
95 |
60 |
121 |
14 |
143 |
12 |
97 |
38 |
119 |
36 |
|
71 |
84 |
49 |
86 |
23 |
132 |
1 |
134 |
47 |
108 |
25 |
110 |
|
50 |
85 |
72 |
83 |
2 |
133 |
24 |
131 |
26 |
109 |
48 |
107 |
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92 |
55 |
78 |
65 |
140 |
7 |
126 |
17 |
116 |
31 |
102 |
41 |
|
77 |
66 |
91 |
56 |
125 |
18 |
139 |
8 |
101 |
42 |
115 |
32 |
|
67 |
80 |
53 |
90 |
19 |
128 |
5 |
138 |
43 |
104 |
29 |
114 |
|
54 |
89 |
68 |
79 |
6 |
137 |
20 |
127 |
30 |
113 |
44 |
103 |
Within this square lie hidden magic squares.
The magic sum of the whole square is 870.
The magic sum of the 4th order squares is 290.
Are there any 8th-order magic squares to count?
Are any of the squares pandiagonal?
Did you remember the 12th-order square itself?
or see the answer
here. |
Here is an Inlaid Magic Square Poster.
Print it out for your classroom if you wish.

A Bimagic Cube of Order 25
Introduction
On the 9th day of
June 2000, the World’s first Bimagic cube was published. I had been trying
various ideas for some time with no success. Then, suddenly I would try
something a bit different by using a bimagic square as a reference and a
“finer-grained” coordinate system and the addition of a new z- direction. Sixty
years of experience has taught me that it is sometime easier to make a large
magic square than a small one and this could be true for cubes too. Besides, I
was getting nowhere with the smaller ones anyway.
The Magic Sum for rows, columns, pillars and the four
major triagonals is given by S1 and for order m=25 by Equation
(1).
S1 = m( 1+m3)/2 =
195,325
…..(1)
Now, if you square all the numbers in the cube and thereby obtain a new magic
cube, the various required sums will be given by Equation (2) instead.
S2 = m(1+m3)(1+2m3)/6
= 2,034,700,525 …..(2)
The Coordinate System
Most people use the decimal number system. Some people might
think that we could use the number system base 25 as it might be easier to
reduce the problem. However, I felt that using a quinary number system on an
order 25 cube will limit the digits from zero to four which might be more
manageable. So,
X = mx2
+ x1 …..(3)
Y = my2
+ y1 …..(4)
Z = mz2
+ z1
…..(5)
Where m=5 a coordinate position, or location. (X, Y, Z) becomes
(x2,
x1, y2, y1, z2, z1)
Now, for each coordinate location, you can assign a Number N given by
N = 55D5
+54D4 + 53D3 + 52D2
+ 51D1 + 1
Where the digits D are found using the following modular equations
D5 =
x2 + x1 + 2y2 + z2 + 2z1
+ 1 (mod 5) ….(6)
D4 =
x2 + 2x1 + 2y1 + 3z2 + 2z1
(mod 5) ….(7)
D3 =
x2 + 3x1 + 3y2 + 2y1 + 3z2
+ 2 (mod 5) ….(8)
D2 =
x2 + 4x1 + 4y2 + 2y1 + 3z1
+ 2 (mod 5) ….(9)
D1 =
x1 + 4y2 + 3y1 + 3z2 + 3z1
(mod 5) …(10)
D0 =
x2 + y2 + y1 + z2 + 2z1
(mod 5)
…..(11)
These simultaneous equations can be solved in reverse to yield the
coordinates, given the number.


Welcome
Books
Bibliography
Plane Geometry
Diagonal Intersections
Probability
Tesseracts
Inlaid Magic Tesseract
Perfect Magic Tesseract
School Material
solution

Last updated
Monday November 02, 2009
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