### Khajuraho method

Use the famous Khajuraho 4x4 panmagic square to construct larger magic squares which are a multiple of 4 (= 8x8, 12x12, 16x16, 20x20, … magic square).

Rewrite the Khajuraho magic square as follows:

Khajuraho magic square                Basic magic square

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

To construct an 12x12 panmagic square, you need the basic square and 8 extending magic squares:

 7 h-4 1 h-2 8 -8 8 -8 16 -16 16 -16 2 h-3 8 h-5 8 -8 8 -8 16 -16 16 -16 h 3 h-6 5 -8 8 -8 8 -16 16 -16 16 h-7 6 h-1 4 -8 8 -8 8 -16 16 -16 16 24 -24 24 -24 32 -32 32 -32 40 -40 40 -40 24 -24 24 -24 32 -32 32 -32 40 -40 40 -40 -24 24 -24 24 -32 32 -32 32 -40 40 -40 40 -24 24 -24 24 -32 32 -32 32 -40 40 -40 40 48 -48 48 -48 56 -56 56 -56 64 -64 64 -64 48 -48 48 -48 56 -56 56 -56 64 -64 64 -64 -48 48 -48 48 -56 56 -56 56 -64 64 -64 64 -48 48 -48 48 -56 56 -56 56 -64 64 -64 64

The highest number in the 12x12 square is 144. Fill in 144 for h and calculate all the numbers. You get the following 12x12 panmagic square.

Panmagic 12x12 square

 7 140 1 142 15 132 9 134 23 124 17 126 2 141 8 139 10 133 16 131 18 125 24 123 144 3 138 5 136 11 130 13 128 19 122 21 137 6 143 4 129 14 135 12 121 22 127 20 31 116 25 118 39 108 33 110 47 100 41 102 26 117 32 115 34 109 40 107 42 101 48 99 120 27 114 29 112 35 106 37 104 43 98 45 113 30 119 28 105 38 111 36 97 46 103 44 55 92 49 94 63 84 57 86 71 76 65 78 50 93 56 91 58 85 64 83 66 77 72 75 96 51 90 53 88 59 82 61 80 67 74 69 89 54 95 52 81 62 87 60 73 70 79 68

This magic square is almost most perfect. Only not all 2x2 sub-squares give 1/3 of the magic sum (1/3 x 870 = 290). Swap the marked numbers and you get the following most perfect magic 12x12 square:

 23 140 1 126 15 132 9 134 7 124 17 142 2 125 24 139 10 133 16 131 18 141 8 123 144 19 122 5 136 11 130 13 128 3 138 21 121 6 143 20 129 14 135 12 137 22 127 4 47 116 25 102 39 108 33 110 31 100 41 118 26 101 48 115 34 109 40 107 42 117 32 99 120 43 98 29 112 35 106 37 104 27 114 45 97 30 119 44 105 38 111 36 113 46 103 28 71 92 49 78 63 84 57 86 55 76 65 94 50 77 72 91 58 85 64 83 66 93 56 75 96 67 74 53 88 59 82 61 80 51 90 69 73 54 95 68 81 62 87 60 89 70 79 52

Use the Khajuraho method to construct magic squares of order is multiple of 4 from 8x8 to infinity. See 8x812x1216x1620x2024x2428x28 and 32x32

12x12, Khajuraho method.xls