### Composite, Matroesjka

How to construct a 12x12 composite magic square?

See in "Scripta Mathematica", 1938, Royal Vale Heath how to construct a 12x12 composite magic square:

• First grid is a 3x3 ‘blown up’ panmagic 4x4 square.
• Second grid is 8x a 3x3 magic square (yellow marked) and 8x the same 3x3 magic square, which is turned upside down (red marked).
• Take 1x number from first grid and add [number minus 1] x 16 from the same cell of the second grid.

1x number from grid with 3x3 'blown up' panmagic 4x4 square

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

+ [number minus 1] x 16 from grid with 3x3 (and upside down) magic square

 6 1 8 4 9 2 6 1 8 4 9 2 7 5 3 3 5 7 7 5 3 3 5 7 2 9 4 8 1 6 2 9 4 8 1 6 4 9 2 6 1 8 4 9 2 6 1 8 3 5 7 7 5 3 3 5 7 7 5 3 8 1 6 2 9 4 8 1 6 2 9 4 4 9 2 6 1 8 4 9 2 6 1 8 3 5 7 7 5 3 3 5 7 7 5 3 8 1 6 2 9 4 8 1 6 2 9 4 6 1 8 4 9 2 6 1 8 4 9 2 7 5 3 3 5 7 7 5 3 3 5 7 2 9 4 8 1 6 2 9 4 8 1 6

= panmagic 12x12 square (consisting of 16 magic 3x3 squares)

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

What are the special magic features of this 12x12 magic square?

(1st) The 12x12 magic square is panmagic and consists of 16 (not proportional) magic 3x3 squares;

(2nd) 9 (proportional) panmagic 4x4 squares are hidden in the 12x12 magic square; see below.

12x12 magic square --> 9x panmagic 4x4 square

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

For example, combine all yellow marked numbers.

= One of the 9 hidden panmagic 4x4 squares:

 290 290 290 290 290 290 290 81 56 93 60 290 62 91 50 87 290 290 290 52 85 64 89 290 290 290 95 58 83 54 290 290

(3th) 27 (proportional) panmagic 8x8 squares are hidden in the 12x12 magic square. Combine 4x number from the same cells of each 3x3 sub-square.

 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

For example, choose 12 and you get the following 8x8 panmagic square.

= One of the 27 panmagic 8x8 squares:

 580 580 580 580 580 580 580 580 580 580 580 81 33 56 104 93 45 60 108 580 17 49 120 88 29 61 124 92 580 580 580 62 110 91 43 50 98 87 39 580 580 580 126 94 27 59 114 82 23 55 580 580 580 52 100 85 37 64 112 89 41 580 580 580 116 84 21 53 128 96 25 57 580 580 580 95 47 58 106 83 35 54 102 580 580 580 31 63 122 90 19 51 118 86 580 580

Use this method to construct magic squares which are a multiple of 4 from 12x12 to infinite. See

12x12, Composite, Matroesjka.xls