1,920 Sub-Magic Squares on 255 4th-Order Magic Tori

Further to my last article (published on the 26th March) Harry White has kindly run a computer programme to verify and validate my hypothesis: The 4th-order magic tori are indeed totally covered by sub-magic 2x2 squares.

As Harry White's results gave 7,712 sub-squares for the 880 Frénicle squares it was necessary to determine which tori had a reduced cover of sub-magic squares. After verification I have found that there are two families of the type 2 and type 5 tori. My latest results and totals are given below. It could be that by using observation and manual input I have overlooked something. Your comments are always welcome.

 

Please note that the type number of this torus is no longer T4.05.2 but now T4.05.3

Conclusion :

Each number of a fourth-order magic torus (or magic square) comes from at least one 2x2 sub-magic square. Or, expressed otherwise, the surface of every fourth-order magic torus is entirely covered by 2x2 sub-magic squares. This hypothesis has since been confirmed by Harry White's computer skills (see above).

Please note that I have published a new article that extends the above findings.

The new table below is a new résumé of the different 4th-order magic tori taking into account the different arrangements of the sub-magic 2x2 squares that cover them.