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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)

 

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.

John Hendricks does mathematics that he likes to do. Sometimes, he comes up with new ideas. You may disagree with him if you wish, That is your right. Although he appears to be known in Magic Squares, etc, he has also done work in probability theory, statistics, and  climatology. 


These pages are an attempt to demonstrate some of these diverse areas covered by John.

Also covered will be selected topics in his specialties of inlaid magic Squares, cubes and tesseracts. For more complete coverage of his work in this arena you may go to

http://www.magic-squares.net/hendricks.htm  or
http://members.shaw.ca/hdhcubes/cube_inlaid.htm or
http://perso.club-internet.fr/cboyer/multimagie/

 

Contents

Plane Geometry

Area of a polygon, circumference of an ellipse, binodal lemniscate, etc.

Diagonals

A discussion concerning diagonal intersections in hypercubes.

Probability

Substance of a talk given to the Annual General Meeting of  the Statistical Association of Manitoba

Tesseracts

A history and explanation. Then pages on Inlaid and Perfect Tesseracts

Books

A description of recent books published by J. R. H. , an order form and a bibliography.

Site Map

List all the pages, with relationships and links.

Bimagic cube

An order 25 bimagic cube presented June 9, 2000

Material for Schools

Posters and Documents suitable for high school math use and available for downloading.

John R. Hendricks

Count the number of magic squares

by John R. Hendricks

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

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.

Site Map

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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