Smart Filling

I wish to thank Francis Gaspalou for bringing to my attention a board game question n° J113 "Savant Remplissage (1er épisode)" that was asked on the site of Diophante:

In French, the question reads as follows:

"Prouver qu'il est possible de remplir les 81 cases d' un tableau 9 x 9 avec les entiers de 1 à 81 de sorte que les sommes des nombres contenus dans tous les carrés 3 x 3 sont identiques.

Pour les plus courageux: même question pour le remplissage des 256 cases d'.un tableau 16 x 16 avec les entiers de 1 à 256 et les sommes identiques des nombres contenus dans tous les carrés 4 x 4."

Translated into English, the question is as follows:

Show that it is possible to fill the 81 cells of a 9 x 9 grid with the whole numbers from 1 to 81 so that the sums of the numbers contained in each of the 3 x 3 squares are identical. For the more courageous: Show that it is possible to fill the 256 cells of a 16 x 16 grid with the whole numbers from 1 to 256 so that the sums of the numbers contained in each of the 4 x 4 squares are identical.

The Reply to the Question

Remembering earlier research carried out in 2016 when preparing "Magic Torus Coordinate and Vector Symmetries" I realised that the question could be answered using this modular coordinate equation:

The following pdf files (in English and French versions) show how the equation is used to produce perfect square order N pandiagonal tori that are always entirely covered by N² x [√N x √N] submagic squares: