Trimagic square, 128th-order

A magic square is said trimagic (or 3-multimagic) if it remains magic after each of its elements have been squared, and if it remains magic after each of its elements have been cubed.

The 8th-order Pfeffermann's bimagic square fails as trimagic, since the sums of the cubed cells varies between 526,400 and 555,200 depending on the line. Only 4 rows have the correct sum: 540,800. The first known trimagic square, including the cubed cells, is credited to a Frenchman, Gaston Tarry.

Gaston Tarry (Villefranche de Rouergue 1843 - Le Havre 1913) was born in the Aveyron, a region in the center of France. He studied mathematics at the Lycée Saint-Louis in Paris, and spent his working life in Algeria in the French Financial Administration (Contributions Diverses de l’Administration des Finances). His profession involved him in the manipulation of numbers! Listen to him describe how his passion for magic squares started when he was in his fifties:

« For about three years, a friend of Alger [Brutus Portier], a keen magician, never stopped titillating me in order that I take an interest in magic squares. I disregarded him, because I didn't like what I considered as game of Chinese headache. One day, my magic teacher (forced teacher in spite of me), claimed me that it was not possible to construct diabolic squares (panmagic) of order 3n, with n being not divisible be 3. This astonished me. »

Astonished, Gaston Tarry investigated the problem and, you guessed it, discovered the first 15th-order panmagic square… This started his passion for magic squares. Over time, he wrote a large number of articles in various scientific magazines of the period.

In 1902, Tarry started his retirement. He remained very active because in 1905 Tarry was the first in the world to construct a trimagic square. His square is order-128. The calculations were done by hand, so we will excuse him for not publishing the complete contents of the 16,384 cells! He called his method the «cabalistic condensator », where we find the description in La Mathématique des Jeux by Maurice Kraïtchik. We can't resist the pleasure of quoting you, Gaston Tarry in the text:

« This condensator is a real magic machine charged at the limit. In discharging it, we will obtain magical effects. The discharge can only stay in a good conductive environment, a magic field. »

Astonishing, isn't it?

In June 2002, we have reconstruct the Gaston Tarry's square in order to verify it. In 1905, Tarry had proved that his square was trimagic, but had computed the first column only. We have completely computed it: now we can confirm that the square is really trimagic!

Download the 128th-order trimagic square of Gaston Tarry, zipped Excel file 527Kb.

For others works of Gaston Tarry, see the end of the page on Sudokus and bimagic squares

A panmagic square (pandiagonal magic, also called diabolic) is a square wich is magic for all its lines, all its columns, and all its full or broken diagonals (not only the two main diagonals).