Bimagic and trimagic squares, from 12th to 16th-order

12th-order bimagic and trimagic squares?


The 12th-order trimagic square constructed in 2002 by Walter Trump, Germany, is of course also a bimagic square. No 12th-order bimagic square was known before.

It means that this Trump square is both:

In September 2007, Pan Fengchu, China, constructed several 12th-order bimagic squares (but not trimagic). For example this one:

13th-order bimagic and trimagic squares?

Bimagic known. Trimagic unknown.

Chen Qinwu and Chen Mutian

2006, February 6th: the first known 13th-order bimagic square is due to Chen Qinwu and Chen Mutian, professors at the Computer Science Department of the Shantou University, Guangdong Province, China (later published in The American Mathematical Monthly, October 2007, page 702).

Incredible: only 3 days later, and completely independently, Jacques Guéron, French teacher of mathematics, constructed another 13th-order bimagic square!

In his Excel file, he explains his construction secrets.

Then, one month later, Jacques Guéron constructed the first 17th and 19th-order bimagic squares...

Nobody has successfully constructed a 13th-order trimagic square. But Fredrik Jansson, Finland, the author of the first known 10th and 11th-order bimagic squares, has done some computing research, using a method I sent him. His best result in May 2006 was this square. It proves at least that it is possible to arrange the 169 numbers in 13 trimagic rows, which was not obvious before.

Simply reordering the rows (first column becoming 85 90 145 118 80 31 139 52 25 57 1 169 113), Nicolas Rouanet remarked in March 2018 that the above Jansson's square has a trimagic diagonal! (85 30 43 68 140 155 15 102 127 11 81 89 159). And he constructed a better square with two more trimagic columns, and again one trimagic diagonal. This square is associative (every pair of numbers symmetrically opposite to the center sum up to the same value, here 170). Who will obtain a square with 13 trimagic columns? If his computing is correct, there is no 13th-order associative trimagic square.

14th-order bimagic and trimagic squares?

Bimagic known. Trimagic impossible.

G. Pfeffermann, France, has constructed (published in 1894 by Commandant Coccoz, AFAS) this 14th-order non-normal bimagic square, "non-normal" meaning using non-consecutive numbers.

In December 2005, Jacques Guéron, constructed this 14th-order nearly bimagic square (using consecutive integers): both diagonals are magic, but only one is bimagic.

In January 2006, using the above square from this webpage, using its good diagonal and its 14 columns (for example, look at the same numbers used in the blue column), but reorganizing cells to get different rows, Chen Qinwu succeeded in constructing the first known 14th-order bimagic square. We can attribute this new square, and they agree, to both Chen Qinwu and Jacques Guéron!

14th-order trimagic series cannot exist, implying that 14th-order trimagic squares are impossible. See the Word document written in 2002 by Walter Trump.

15th-order bimagic and trimagic squares?

Bimagic known. Trimagic unknown.

Gaston Tarry, France, published three different 15th-order "nearly" bimagic squares using consecutive numbers:

His best one is the last one: the 15 rows are bimagic, the 15 columns are bimagic, but the 2 diagonals are "only" magic.

In 2003, I was pleased to improve his square and to obtain the following Tarry-Boyer square: one diagonal is magic, but the other one is now bimagic.

And in January 2006, Chen Qinwu constructed the first known 15th-order bimagic square.

16th-order bimagic and trimagic squares?


The first known 16th-order bimagic square seems to be the following one, published by Gaston Tarry, France, in the Revue Scientifique, 1903. And it is a partial trimagic square: the 16 rows are trimagic.

David Collison, USA, in 1991, and Jacques Guéron, France, in 2002, have succeeded in constructing 16th-order non-normal trimagic squares.

The first known 16th-order normal trimagic square is due to Chen Mutian and Chen Qinwu. The square was found in May 2005. It is "symmetrical" left/right: the ith element of a row + the (17-i)th element of this same row = 16² + 1 = 257. In March 2008, Li Wen constructed another 16th-order trimagic square.

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