Story of the smallest trimagic square
Text written by Walter Trump, in January 2003.


On April 25th 2002 a friend of mine from Holland, Aale de Winkel, showed me a new site with the web address www.multimagie.com. The Frenchman Christian Boyer describes on this site the historical development of multimagic squares and the discovery of the first tetra- and penta-magic squares by himself and by his friend André Viricel. A multimagic square is a magic square remaining magic when all its numbers are raised to various powers (squared, cubed, …).

In order to construct multimagic squares of order n, it’s interesting to calculate multimagic series of order n: all possible combinations of n different, natural numbers not greater than n², where the sum of the first, second, ... power corresponded to the particular constant. In other words, among all sequences of numbers these are the potential candidates for the rows and columns of multimagic squares.

A trimagic series of order 8.

1  + 19  + 23  + 24  + 36  + 42  + 53  + 62  =     260 = (1  + 2  + ... + 64 ) / 8
1
2 + 192 + 232 + 242 + 362 + 422 + 532 + 622 =  11,180 = (12 + 22 + ... + 642) / 8
1
3 + 193 + 233 + 243 + 363 + 423 + 533 + 623 = 540,800 = (13 + 23 + ... + 643) / 8

A busy e-mail conversation between Christian Boyer and me developed about bimagic series. Mid-may, he had the idea to switch from bimagic series to TRImagic series, asking me to check his number of 121 trimagic series of order 8. I confirmed this number May 18th. It’s the true starting point of my trimagic search.

We discussed the question whether there is a smaller trimagic square than the one found by Benson which is of order 32 (which means 32 rows and 32 columns). This discussion was accompanied, even though less intensively, by other amateurs of magic squares from all over the world. Some of them assumed that the number 32 already was the smallest possible order (e.g. George Chen from Taiwan), others thought it was the number 27 (André Viricel from France), none of them took orders smaller than 16 into account.

Christian Boyer had already proved that orders smaller than 10 were out of the question. I could confirm his results by means of other methods and I was additionally able to prove that a trimagic square of a smaller order than the order 12 does not exist.

Within these reflections we calculated - for small orders n- all possible combinations. Up to the order 10 the trimagic rows could be found relatively fast. With the order 11, I came to the same result as Christian Boyer – after I had corrected a small mistake. In order to find all trimagic series of order 12, some days of computing (with an ordinary PC) were necessary. I found not less than 2,226,896  trimagic series of order 12. So far this result has not been confirmed by anyone. All series consist of 6 odd and 6 even integers.

Trimagic series, with 6 odd and 6 even integers.

 1   5  45  69  73 143  70  72  74  86  90 142
 1   5  45  75  91 143  68  70  72  74  84 142
 1   5  47  77  87 143  60  70  72  82  84 142
  ...
93  95 101 111 125 139   4  22  40  42  48  50
93  97 113 115 125 129  10  26  30  42  44  46
99 111 113 115 117 125  22  26  28  30  40  44

Now the question arose whether you can build with the trimagic series a trimagic square of order 12 after all. Imagine that the period of time afterwards was like going on a treasure hunt. Where to search? How to search? Does the golden treasure really exist?

Would you be able to be successful by means of 'Brute Force'? With the PC one would be able to calculate normal magic squares of order 12 and find out if there are additional trimagic properties. Such an attempt ended in a disaster because of the incredibly high number of magic squares of order 12. My statistics approach showed that the number of these squares is higher than 10188 (a number with 189 digits – beyond any human imagination).

Another approach could be to find 12 independent trimagic series, which form the rows of the square. Afterwards one could try to change and rechange the numbers within the rows until the columns, too, end up in trimagic series. But, this procedure doesn´t seem to be very promising either because the computing time would probably range within billions of years.

The „claim“ for the treasure hunt still had to be fenced even narrower. That is why the search was limited to so called axially symmetrical squares. With these squares the sum of two numbers in symmetrical cells is always n²+1. Two numbers with this sum are called complementary. Now, for the rows, only symmetrical trimagic series were important and for the columns such without complementary pairs of numbers. Besides, only the left half of the square (72 numbers) had to be figured out. The right half resulted from symmetry.

Thus, the „claim“ for the treasure hunt was found, but no yet the method how to search. Searching for gold one can check out first of all whether there are any metals at all to be found. Searching for magic squares one can look for a start for so called semi-magic squares. With these squares the sum of numbers in the diagonals doesn´t play a role.

With semi-magic squares the characteristics remain when you change the rows and columns. Thus, one can put any number into the left-top case. I chose the number 23, because it occurs most frequently within trimagic series. So you have only 3,646 possibilities for the first row, 107,099 possibilities for the first columns, 33,004 possibilities for any further row and 473,663 for any further column. There are still too many possibilities for a strategy of trial and error. Having defined the first row and the first column, there are only 55 numbers left to be calculated. With a trick the potential trimagic rows and columns can be described by only one 64-bit-integer each. With the help of AND-Function one can find out extremely fast if two rows have numbers in common. First, five other independent columns are added to the first column. Hereby a so-called backtracking procedure is used. That means, one can follow all conceivable roads, but, if one ends up in a dead-end-street one goes back to the last crossroads.  The rows, too, are searched by means of a backtracking procedure. Hereby, not only the independence of rows must be respected, but, additionally, every row must have in common with every column one number.

For the algorithm I made use of the programming language GB32, which was developed in Germany by the firm Gfa-Soft, all in all by Frank Ostrowski. GB32 can be used effectively to solve mathematical problems, although it is not very popular yet unfortunately. The first version of my program was completed in May 2002. Some hours of computing resulted in a partial success only. Several „incomplete“ trimagic squares , where only 4 columns didn´t fulfil the trimagic demands, were found. On June 1st, a nearly complete semi-trimagic square was found, where all 12 rows and 10 columns were trimagic.

Nearly semi-trimagic square (June 1st, 2002).

53

67

74

69

1

3

142

144

76

71

78

92

79

72

64

5

35

7

138

110

140

81

73

66

118

11

9

55

85

68

77

60

90

136

134

27

15

17

13

82

62

57

88

83

63

132

128

130

124

108

122

126

43

56

89

102

19

23

37

21

129

46

36

120

104

29

116

41

25

109

99

16

38

24

34

47

114

18

127

31

98

111

121

107

10

133

95

59

96

33

112

49

86

50

12

135

48

131

75

8

40

39

106

105

137

70

14

97

44

113

139

100

30

42

103

115

45

6

32

101

93

94

125

141

117

65

80

28

4

20

51

52

119

54

84

58

143

22

123

2

87

61

91

26

Christian Boyer answered consequently:

Besides, Christian Boyer changed the above-mentioned square in a way, that all columns were at least bimagic and that the diagonals resulted in the normal magic sum. Thus, a magic square, which was semi-bimagic and almost trimagic.

Semi-bimagic and almost trimagic transformation by Christian Boyer.

53

69

71

78

1

142

3

144

67

74

76

92

79

5

81

73

35

138

7

110

72

64

140

66

118

55

136

134

85

68

77

60

11

9

90

27

15

82

132

128

62

57

88

83

17

13

63

130

124

126

23

37

43

89

56

102

108

122

19

21

129

120

109

99

104

29

116

41

46

36

25

16

38

47

111

121

114

18

127

31

24

34

98

107

10

59

50

12

96

33

112

49

133

95

86

135

48

8

70

14

40

106

39

105

131

75

137

97

44

100

6

32

30

103

42

115

113

139

45

101

93

141

20

51

117

65

80

28

94

125

4

52

119

58

61

91

143

22

123

2

54

84

87

26

In spite of this partial success the program turned out as not be working effectively enough. With that version it would have probably taken years until at least a semi-trimagic square could have been found.

Some ideas how to improve the program were haunting me and took shape in the night of Saturday, June 8th 2002. I got up round 6 o´clock and added my new ideas to the existing program. Round 9 o´clock already the first semi-trimagic squares of order 12 were found.

The first “metal samples” had been found. Now one had to find out whether there was gold among it. Another computer program was supposed to help to create trimagic diagonals by changing the rows and columns. After some hours, exactly with the 88th semi-trimagic square, a trimagic diagonal could be constructed and – for reasons of symmetry – the second diagonal was automatically trimagic, too. But I assumed that there must have been a mistake as the success had come about too quickly that day. So I checked all the demanded characteristics additionally with the help of a spreadsheet program. Every number between 1 and 144 was exactly used once, all the sums were correct.

The original trimagic square found by Walter Trump in June 2002.

1

22

33

41

62

66

79

83

104

112

123

144

9

119

45

115

107

93

52

38

30

100

26

136

75

141

35

48

57

14

131

88

97

110

4

70

74

8

106

49

12

43

102

133

96

39

137

71

140

101

124

42

60

37

108

85

103

21

44

5

122

76

142

86

67

126

19

78

59

3

69

23

55

27

95

135

130

89

56

15

10

50

118

90

132

117

68

91

11

99

46

134

54

77

28

13

73

64

2

121

109

32

113

36

24

143

81

72

58

98

84

116

138

16

129

7

29

61

47

87

80

34

105

6

92

127

18

53

139

40

111

65

51

63

31

20

25

128

17

120

125

114

82

94

The treasure had been found! A trimagic square of order 12 was shown on the screen. The search for the smallest trimagic square had been successful.

The day after I sent the number square by e-mail to all the amateurs of magic squares I knew and asked them to check the result. Some answers can be found and read in the attachment. All confirmed that the result was correct.

My special thanks go to Christian Boyer (France), who triggered off the search by his publications and his ideas. Without his own contributions and his critical examination of my calculation, the search wouldn´t have been successful. My special thanks go to Aale de Winkel (Holland), who encouraged me during the year again and again to go on working in the field of magic squares. Harvey Heinz (Canada) and George Chen (Taiwan) gave me a big helping hand, too.

Walter Trump, Nürnberg, January 2003.


E-mail answers on behalf of the discovery of the smallest trimagic square

Christian Boyer (France, ancient technical director of Microsoft France, founder of a software firm, inventor of the first tetra- and penta-magic squares,  Webpublisher: www.multimagie.com)

Yung C. Chen, alias George Chen (Taiwan, inventor of construction methods, author of essays dealing with combination)

Aale de Winkel (The Netherlands, physicist and information scientist, publishes the  magic square encyclopedia.)

Harvey Heinz (Canada, author of Magic Square Lexicon, sponsors worldwide the biggest website on magic squares)


Return to the home page http://www.multimagie.com