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.

4th-order panmagic or pandiagonal torus type 4.01


order 4 sub-magic 2x2 squares panmagic torus type T4.01 (here T4.01.1) the torus
4th-order panmagic or pandiagonal torus type 4.01
represented by Frénicle index 102, Dudeney type I

order 4 sub-magic 2x2 squares panmagic torus type T4.01 (here T4.01.1) the 2x2 squares
4th-order panmagic or pandiagonal torus type 4.01
represented by 16 sub-magic 2x2 squares

4th-order semi-panmagic or semi-pandiagonal torus type 4.02.1


order 4 sub-magic 2x2 squares semi-pandiagonal magic torus type T4.02.1 now T4.02.1.01 the torus
4th-order semi-panmagic or semi-pandiagonal torus type 4.02.1
represented by Frénicle index 016, Dudeney type VI

order 4 sub-magic 2x2 squares semi-pandiagonal magic torus type T4.02.1 now T4.02.1.01 the 2x2 squares
4th-order panmagic or pandiagonal torus type 4.02.1
represented by 12 sub-magic 2x2 squares

4th-order semi-panmagic or semi-pandiagonal torus type 4.02.2


order 4 sub-magic 2x2 squares semi-pandiagonal magic torus type T4.02.2 now T4.02.2.01 the torus
4th-order semi-panmagic or semi-pandiagonal torus type 4.02.2
represented by Frénicle index 021, Dudeney type II

order 4 sub-magic 2x2 squares semi-pandiagonal magic torus type T4.02.2 now T4.02.2.01 the 2x2 squares
4th-order semi-panmagic or semi-pandiagonal torus type 4.02.2
represented by 8 sub-magic 2x2 squares

4th-order partially panmagic or partially pandiagonal torus type 4.03


order 4 sub-magic 2x2 squares partially panmagic torus type T4.03 now T4.03.1.1 the torus
4th-order partially panmagic or partially pandiagonal torus type 4.03
represented by Frénicle index 046, Dudeney type VI "Simple"

order 4 sub-magic 2x2 squares partially panmagic torus type T4.03 now T4.03.1.1 the 2x2 squares
4th-order partially panmagic or partially pandiagonal torus type 4.03
represented by 8 sub-magic 2x2 squares

4th-order partially panmagic or partially pandiagonal torus type 4.04


order 4 sub-magic 2x2 squares partially panmagic torus type T4.04 now T4.04.1 the torus
4th-order partially panmagic or partially pandiagonal torus type 4.04
represented by Frénicle index 040, Dudeney type VIII "Simple"

order 4 sub-magic 2x2 squares partially panmagic torus type T4.04 now T4.04.1 the 2x2 squares
4th-order partially panmagic or partially pandiagonal torus type 4.04
represented by 4 sub-magic 2x2 squares

4th-order basic magic torus type 4.05.1


order 4 sub-magic 2x2 squares basic magic torus type T4.05.1 now T4.05.1.02 the torus
4th-order basic magic torus type 4.05.1
represented by Frénicle index 002, Dudeney type VI Simple

order 4 sub-magic 2x2 squares basic magic torus type T4.05.1 now T4.05.1.02 the 2x2 squares
4th-order basic magic torus type 4.05.1
represented by 8 sub-magic 2x2 squares

4th-order basic magic torus type 4.05.2


order 4 sub-magic 2x2 squares basic magic torus type T4.05.2 now T4.05.3.1 the torus
4th-order basic magic torus type 4.05.2
represented by Frénicle index 003, Dudeney type XII Simple

order 4 sub-magic 2x2 squares basic magic torus type T4.05.2 now T4.05.3.1 the 2x2 squares
4th-order basic magic torus type 4.05.2
represented by 4 sub-magic 2x2 squares


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 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

No comments:

Post a Comment

Or, should you prefer to send a private message, please email william.walkington@wandoo.fr