Three Small Magic Squares

Wednesday, October 18, 2017

Structure of the Dürer magic square

16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1

15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .

Base 4 —

33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .

Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays —

3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .

Base 2 –

1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .

Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —

1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .

— Steven H. Cullinane,
October 18, 2017

See also recent related analyses of
noted 3×3 and 5×5 magic squares.

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"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"

Related material —

"Inner Truth" in this journal
"On Linguistic Creation," June 25, 1999
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .

See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —

— and Conway-Norton-Ryba in this journal.

Some remarks on an order-five magic square over GF(5²):

on the numbers 0 to 24:

22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8

Base-5:

42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13

Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(5²)) …

All vector row sums = (0, 0) (or 0, over GF(5²)).
All vector column sums = same.

Above array as two
orthogonal Latin squares:

4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3

— Steven H. Cullinane,
October 16, 2017

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