### The 3 basic 4x4 panmagic squares

48 of the 880 pure magic 4x4 squares are panmagic (= group 1). These squares have (just like the larger most perfect magic squares) the following structure:

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

The sum of two numbers of the same colour is each time (the lowest plus the highest number of the magic square: 1+16=) 17.

You need to know only the following panmagic 4x4 basic squares, to construct all 48 panmagic 4x4 squares (excluding rotation and/or mirroring):

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

Make a 2x2 carpet of one of the basic squares and you can get all 16 (x 3 = 48) squares by shifting on the carpet. See for example:

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

You can get 8 results in stead of 1 result of the above yellow marked square by rotating and/or mirroring:

 yellow marked square 4 14 11 5 Mirroring 5 11 14 4 15 1 8 10 10 8 1 15 6 12 13 3 3 13 12 6 9 7 2 16 16 2 7 9 rotation by 1 quarter 9 6 15 4 Mirroring 4 15 6 9 7 12 1 14 14 1 12 7 2 13 8 11 11 8 13 2 16 3 10 5 5 10 3 16 rotation by 2 quarters 16 2 7 9 Mirroring 9 7 2 16 3 13 12 6 6 12 13 3 10 8 1 15 15 1 8 10 5 11 14 4 4 14 11 5 rotation by 3 quarters 5 10 3 16 Mirroring 16 3 10 5 11 8 13 2 2 13 8 11 14 1 12 7 7 12 1 14 4 15 6 9 9 6 15 4

There are 3 basic 4x4 panmagic squares. There are 16 possibilities by shifting on the carpet. There are 8 possibilities by rotating and/or mirroring. This gives in total 3 x 16 x 8 is 384 possibilities (including rotating and/or mirroring).

4x4, the 3 basic panmagic 4x4 squares.xl