### Symmetric transformation (Liki)

Paulus Gerdes introduced the Liki magic square (see http://plus.maths.org/content/new-designs-africa). He showed that it is possible to transform a square with consecutive numbers into a magic square by swapping half of the numbers symmetrically. You can use this method to construct magic squares which are a multiple of 4 (= 4x4, 8x8, 12x12, 16x16, ... magic square).

Paulus Gerdes constructed the following symmetric 8x8 magic square:

8x8 square with consecutive numbers

 232 240 248 256 264 272 280 288 260 260 36 1 2 3 4 5 6 7 8 100 9 10 11 12 13 14 15 16 164 17 18 19 20 21 22 23 24 228 25 26 27 28 29 30 31 32 292 33 34 35 36 37 38 39 40 356 41 42 43 44 45 46 47 48 420 49 50 51 52 53 54 55 56 484 57 58 59 60 61 62 63 64

Symmetric 8x8 magic square

 260 260 260 260 260 260 260 260 260 260 260 1 63 3 61 60 6 58 8 260 56 55 11 12 13 14 50 49 260 17 18 46 45 44 43 23 24 260 40 26 38 28 29 35 31 33 260 32 34 30 36 37 27 39 25 260 41 42 22 21 20 19 47 48 260 16 15 51 52 53 54 10 9 260 57 7 59 5 4 62 2 64

It is a beautiful result, but it is possible to get a more magic result.

 52 56 60 64 68 72 76 80 180 184 188 192 196 200 204 208 260 260 10 26 1 2 3 4 5 6 7 8 42 58 9 10 11 12 13 14 15 16 74 90 17 18 19 20 21 22 23 24 106 122 25 26 27 28 29 30 31 32 138 154 33 34 35 36 37 38 39 40 170 186 41 42 43 44 45 46 47 48 202 218 49 50 51 52 53 54 55 56 234 250 57 58 59 60 61 62 63 64

 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 260 260 130 130 1 63 62 4 5 59 58 8 130 130 56 10 11 53 52 14 15 49 130 130 48 18 19 45 44 22 23 41 130 130 25 39 38 28 29 35 34 32 130 130 33 31 30 36 37 27 26 40 130 130 24 42 43 21 20 46 47 17 130 130 16 50 51 13 12 54 55 9 130 130 57 7 6 60 61 3 2 64

Each 1/2 row/column gives 1/2 of the magic sum.

By refining the grids you can construct an ultramagic 8x8 square.

1x number from first grid                     +       1x [number -/- 1] from second grid   =

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

symmetric transformation                    =      Ultra magic 8x8 square

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

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

Use this method to construct magic squares of order is multiple of 4 from 4x4 to infinity. See 4x48x812x1216x1620x2024x2428x2832x32

8x8, Symmetric transformation (Liki).xls