### Composite 9x9 magic square (2)

Use 9 proportional (semi)magic 3x3 squares to construct a 9x9 magic square. Proportional means that all 9 (semi)magic 3x3 squares have the same magic sum of (1/3 x 369 = ) 123. Use the row and column coordinates of the 3x3 magic square. Don't use as row coordinates the numbers 0 up to 2, but use the numbers 1 up to (9x3 = ) 27 instead. To get the numbers proportional divided, use the following table:

 1 18 23 42 2 16 24 42 3 17 22 42 4 12 26 42 5 10 27 42 6 11 25 42 7 15 20 42 8 13 21 42 9 14 19 42

Construct the 9 (semi)magic 3x3 squares.

Row coordinate +27x column coordinate = (semi)magic 3x3 square

 18 1 23 0 2 1 18 55 50 23 18 1 2 1 0 77 45 1 1 23 18 1 0 2 28 23 72 16 2 24 0 2 1 16 56 51 24 16 2 2 1 0 78 43 2 2 24 16 1 0 2 29 24 70 17 3 22 0 2 1 17 57 49 22 17 3 2 1 0 76 44 3 3 22 17 1 0 2 30 22 71 12 4 26 0 2 1 12 58 53 26 12 4 2 1 0 80 39 4 4 26 12 1 0 2 31 26 66 10 5 27 0 2 1 10 59 54 27 10 5 2 1 0 81 37 5 5 27 10 1 0 2 32 27 64 11 6 25 0 2 1 11 60 52 25 11 6 2 1 0 79 38 6 6 25 11 1 0 2 33 25 65 15 7 20 0 2 1 15 61 47 20 15 7 2 1 0 74 42 7 7 20 15 1 0 2 34 20 69 13 8 21 0 2 1 13 62 48 21 13 8 2 1 0 75 40 8 8 21 13 1 0 2 35 21 67 14 9 19 0 2 1 14 63 46 19 14 9 2 1 0 73 41 9 9 19 14 1 0 2 36 19 68

Put the 9 (semi)magic 3x3 squares together.

9x9 magic square

 18 55 50 16 56 51 17 57 49 77 45 1 78 43 2 76 44 3 28 23 72 29 24 70 30 22 71 12 58 53 10 59 54 11 60 52 80 39 4 81 37 5 79 38 6 31 26 66 32 27 64 33 25 65 15 61 47 13 62 48 14 63 46 74 42 7 75 40 8 73 41 9 34 20 69 35 21 67 36 19 68

Each 1/3 row/column gives 1/3 of the magic square and the 9x9 magic square is 3x3 compact, but not panmagic.

9x9, Composite (2a).xls

Use a 3x9 magic rectangle to get a symmetric result:

 1 25 16 42 2 23 17 42 15 21 6 42 20 18 4 42 19 14 9 42 24 10 8 42 22 7 13 42 11 5 26 42 12 3 27 42

Row coordinate +27x column coordinate = (semi)magic 3x3 square

 25 1 16 0 2 1 25 55 43 16 25 1 2 1 0 70 52 1 1 16 25 1 0 2 28 16 79 23 2 17 0 2 1 23 56 44 17 23 2 2 1 0 71 50 2 2 17 23 1 0 2 29 17 77 21 15 6 0 2 1 21 69 33 6 21 15 2 1 0 60 48 15 15 6 21 1 0 2 42 6 75 18 20 4 0 2 1 18 74 31 4 18 20 2 1 0 58 45 20 20 4 18 1 0 2 47 4 72 14 19 9 0 2 1 14 73 36 9 14 19 2 1 0 63 41 19 19 9 14 1 0 2 46 9 68 10 24 8 0 2 1 10 78 35 8 10 24 2 1 0 62 37 24 24 8 10 1 0 2 51 8 64 7 22 13 0 2 1 7 76 40 13 7 22 2 1 0 67 34 22 22 13 7 1 0 2 49 13 61 5 11 26 0 2 1 5 65 53 26 5 11 2 1 0 80 32 11 11 26 5 1 0 2 38 26 59 3 12 27 0 2 1 3 66 54 27 3 12 2 1 0 81 30 12 12 27 3 1 0 2 39 27 57

9x9 magic square

 25 55 43 23 56 44 21 69 33 70 52 1 71 50 2 60 48 15 28 16 79 29 17 77 42 6 75 18 74 31 14 73 36 10 78 35 58 45 20 63 41 19 62 37 24 47 4 72 46 9 68 51 8 64 7 76 40 5 65 53 3 66 54 67 34 22 80 32 11 81 30 12 49 13 61 38 26 59 39 27 57

Each 1/3 row/column gives 1/3 of the magic square and the 9x9 magic square is symmetric (but is not [fully] 3x3 compact and not panmagic).

9x9, Composite (2b).xls