Thursday, 28 March 2013

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.

order 4 sub-magic 2x2 squares panmagic torus type T4.01 now T4.01.1


order 4 sub-magic 2x2 squares semi-pandiagonal magic torus type T4.02.1 now T4.02.1.01


order 4 sub-magic 2x2 squares semi-pandiagonal magic torus type T4.02.2 now T4.02.2.01


order 4 sub-magic 2x2 squares partially panmagic torus type T4.03 now T4.03.1.1


order 4 sub-magic 2x2 squares partially panmagic torus type T4.04 now T4.04.1


order 4 sub-magic 2x2 squares basic magic torus type T4.05.1 now T4.05.1.02


order 4 sub-magic 2x2 squares basic magic torus type T4.05.2 now T4.05.3.1
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).

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.

4th-order magic torus or magic tori summary order 4

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