Panmagic square

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A panmagic square, pandiagonal magic square, diabolic square, diabolical square or diabolical magic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

A panmagic square remains a panmagic square not only under rotation or reflection, but also if a row or column is moved from one side of the square to the opposite side. As such, an n×n panmagic square can be regarded as having 8n² orientations.

[edit] 4×4 panmagic squares

The smallest non-trivial panmagic squares are 4×4 squares.

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

In 4×4 panmagic squares, the magic constant of 34 can be seen in a number of patterns in addition to the rows, columns and diagonals:

• Any of the sixteen 2×2 squares, including those that wrap around the edges of the whole square, e.g. 14+11+4+5, 1+12+15+6
• The corners of any 3×3 square, e.g. 8+12+5+9
• Any pair of horizontally or vertically adjacent numbers, together with the corresponding pair displaced by a (2, 2) vector, e.g. 1+8+16+9

Thus of the 86 possible sums adding to 34, 52 of them form regular patterns, compared with 10 for an ordinary 4×4 magic square.

There are only three distinct 4×4 panmagic squares, namely the one above and the following:

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

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

In any 4×4 magic square, any two numbers at the opposite corners of a 3×3 square add up to 17. Consequently, no 4×4 panmagic squares are associative.

[edit] 5×5 panmagic squares

There are many 5×5 panmagic squares. Unlike 4×4 panmagic squares, these can be associative. The following is a 5×5 associative panmagic square:

20	8	21	14	2
11	4	17	10	23
7	25	13	1	19
3	16	9	22	15
24	12	5	18	6

[edit] 4×4 magic square with arbitrary sum

M=35.

7	12	1	15
2	14	8	11
17	3	10	5
9	6	16	4

Notice the spot of 13 is incremented to 14 in order to obtain the needed sum. We can get 36 by incrementing at 9, and 37 by incrementing at 5.

This symmetric magic square is known with in Mathematics department prior to 1980 at Sainik School, Amaravathinagar. Origin unknown though.