**Story of the smallest trimagic square**

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.

1 + 19 +
23 + 24 + 36 + 42 + 53 + 62 =
260 = (1 + 2 + ... +
64 ) / 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.

1 5
45 69 73 143 70 72 74 86
90 142 |

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 10^{188} (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.

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:

*Dear Walter, Hope you will be the "first in the world"
to discover a 12th-order trimagic square or at least a semi-trimagic square.
As far as I know, nobody else in the world is working on the subject, so
you have a good chance to win the jackpot! 12 trimagic rows and 10 trimagic
columns (and probably one diagonal) is already an incredible advance on
the subject, better than I have ever imagined only one or two weeks ago.
Christian.*

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.

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.

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)

*Dear Walter, Excellent square! Incredible result! Best than
the famous Benson's square!As you asked, I have checked the properties,
and of course all is OK.And all the integers from 1 to 144 are present
;-)So, you have win the right to be in the record table in my site!Congratulations. Christian Boyer.*

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

*Dear Walter,Congratulation.There is will, there is a way.Your
achievement is marvelous, it is a milestone on magic squares.Best Regards,George*

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

*Dear Walter Congratulations, trimagicness of square I verified,
...... this one is a tremendous achievement indeed.It would be
great to upload the order 12 trimagic squares onto the encyclopedia's database
for future reference (but I assume it might be prudent to publish this find
in scientific journals as well)greetings Aale*

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

*Congratulations Walter, you hit the jackpot.What is especially
significant about this accomplishment is the amount by which you beat the
previous record!Previous record holders only halved the older record.
Your result is one third the older one.Boy, I bet you were excited when
you saw the results of that 88th square! We are all excited with you.Like
Christian, I can't wait to put something on my Web site about this.However,
I wait for the O.K. from you. You will want to publish in a mathematical
journal, I assume. However, this is something for the local papers and the
news networks also. Magic squares is something that the general public can
identify with.Congratulations again, Walter.Regards for nowHarvey*

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