### Composite, Proportional (1)

René Chrétien had noticed the 15x15 composite (4) magic square and showed me it is possible to use the method to construct magic squares of even orders as well.

Construct the 8x8 magic square by using 4 proportional 4x4 panmagic squares. The squares are proportional because all 4 panmagic 4x4 squares have the same magic sum of (1/2 x 260 = ) 130. We use the basic key method (4x4) to produce the panmagic 4x4 squares.  As row coordinates don't use 0 up to 3 but use 0 up to (4x4 -/- 1 = ) 15 instead. Take care that the sum of the row coordinates in each 4x4 square is the same  (0+7+8+15 = 1+6+9+14 = 2+5+10+13 = 3+4+11+12 = 30) to get proportional squares.

1x row coordinate +16x column cordinate + 1 = panmagic 4x4 square

 0 7 8 15 0 3 1 2 1 56 25 48 8 15 0 7 3 0 2 1 57 16 33 24 7 0 15 8 2 1 3 0 40 17 64 9 15 8 7 0 1 2 0 3 32 41 8 49 1 6 9 14 0 3 1 2 2 55 26 47 9 14 1 6 3 0 2 1 58 15 34 23 6 1 14 9 2 1 3 0 39 18 63 10 14 9 6 1 1 2 0 3 31 42 7 50 2 5 10 13 0 3 1 2 3 54 27 46 10 13 2 5 3 0 2 1 59 14 35 22 5 2 13 10 2 1 3 0 38 19 62 11 13 10 5 2 1 2 0 3 30 43 6 51 3 4 11 12 0 3 1 2 4 53 28 45 11 12 3 4 3 0 2 1 60 13 36 21 4 3 12 11 2 1 3 0 37 20 61 12 12 11 4 3 1 2 0 3 29 44 5 52

Put the 4 panmagic 4x4 squares together.

8x8 magic square

 1 56 25 48 2 55 26 47 57 16 33 24 58 15 34 23 40 17 64 9 39 18 63 10 32 41 8 49 31 42 7 50 3 54 27 46 4 53 28 45 59 14 35 22 60 13 36 21 38 19 62 11 37 20 61 12 30 43 6 51 29 44 5 52

The 8x8 magic square is not fully 2x2 compact. Swap numbers (see Khajuraho method) to get a perfect magic square.

Franklin panmagic 8x8 square

 2 56 25 47 1 55 26 48 57 15 34 24 58 16 33 23 40 18 63 9 39 17 64 10 31 41 8 50 32 42 7 49 4 54 27 45 3 53 28 46 59 13 36 22 60 14 35 21 38 20 61 11 37 19 62 12 29 43 6 52 30 44 5 51

This 8x8 magic square is panmagic, 2x2 compact and each 1/2 row/column/diagonal gives 1/2 of the magic sum.

I have used composite method, proportional (1) to construct 8x8, 9x9, 12x12a, 12x12b, 15x15a, 15x15b, 16x16a, 16x16b, 18x18, 20x20a, 20x20b, 21x21a, 21x21b, 24x24a, 24x24b, 24x24c, 27x27a, 27x27b, 28x28a, 28x28b, 30x30a, 30x30b, 32x32a, 32x32b and 32x32c

8x8, Composite, Prop. (1).xls