Bimagic squares

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 German naturalized Frenchman Georges Pfeffermann, built the first bimagic square in 1890. Rather than publishing it completely, he proposed it partially, in the form of a puzzle, in the fortnightly magazine Les Tablettes du Chercheur - Journal des Jeux d'Esprit et de Combinaisons, number 2, January 15, 1891.

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! Hint: the sums of the rows, columns and diagonals of the 8th-order square (n=8) have to equal 260 (=n(n˛+1)/2), and the sums when the numbers are squared have to equal 11,180 (=n(n˛+1)(2n˛+1)/6).

 56 8 18 9 20 48 29 10 26 13 64 4 5 30 12 60 15 63 41 50 55 11 58 45 61 42 27 39 62 37 51 3

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 published articles in Les Tablettes, wrote that this first bimagic square was a « very remarkable square ».

The smallest possible bimagic squares are of the same 8th-order as this one of Pfeffermann.

Pfeffermann 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. For example:

In May 2014, great news: Walter Trump and Francis Gaspalou announced that they have computed the number of 8x8 bimagic squares: