Wednesday, 6 February 2019

Smart Filling

The Board Game Question


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:

A Modular Coordinate Equation for Perfect Square Order Pandiagonal Magic Tori or Magic Squares

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: