The following text is written by Christian Boyer and is
coming from his web site
www.multimagie.com/indexengl.htm, Thanks for his
authorization to use his text.
A magic square is said to be
bimagic (or 2-multimagic) if it remains magic after each of its numbers is
replaced by its square. A frenchman, G. Pfeffermann , built the first bimagic
square in 1890. Rather than publishing it integrally, he proposed it partially,
in the form of a puzzle, in the fortnightly magazine Les Tablettes du Chercheur
- Journal de Jeux d'Esprit et de Combinaisons , number 2 of January 15, 1891.
WANTED! Who was this
G. Pfeffermann ? We have never found any information about him, even though
he authored numerous articles on magic squares published in France, mainly
between 1890 and 1896. Even his full firstname is unknown to us. We have
only found a few times this signature: "Gg. Pfeffermann". Probably Georges.
Or perhaps Grégoire? Already, in 1926, André Gérardin (Nancy) was astonished
about that in the Annales de la Société Scientifique de Bruxelles : " It's a
mathematician about which we have very little bibliographic information,
because few families take care of the scientific memory of their parents or
of the keeping of their archives".
Hope at least that it was not a pseudonym! So if you have some
information, even minimal, about this mysterious Mr G. Pfeffermann, contact
Here is the square as it was published at that time. It is up to you to
complete it with the remainder of the 64 numbers in order to obtain a bimagic
square! A hint: the sums of the rows, columns and diagonals of the 8th-order
square (n=8) have to be equal to 260 (=n(n²+1)/2), and the sums when the numbers
are squared have to be equal to 11,180 (=n(n²+1)(2n²+1)/6).
Pfeffermann published the solution
a fortnight later, in this same magazine. The editorial staff of
the Tablettes extended to the author of this first bimagic square
their most sincere compliments for this real tour de force that
he has just accomplished . And the famous Edouard Lucas (1842-1891),
who was a writer of articles in Les Tablettes , wrote that this
first bimagic square was a very remarkable square .
continued to publish, in the following issues, several other 8th-order bimagic
squares, and also 9th-order bimagic squares . Today we know how to construct, by
various methods, bimagic squares of various orders. And the smallest bimagic
squares possible are very certainly of the same 8th-order as this one of