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.

 1  + 19  + 23  + 24  + 36  + 42  + 53  + 62  =     260 = (1  + 2  + ... + 64 ) / 812 + 192 + 232 + 242 + 362 + 422 + 532 + 622 =  11,180 = (12 + 22 + ... + 642) / 813 + 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.

 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  5093  97 113 115 125 129  10  26  30  42  44  4699 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.

 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

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.