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.
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).