Shift method (2)
Put in the first row of the first grid the numbers 0 up to 8. Construct the second and the third row of the first grid by shifting the first row each time 3 places to the left.
First grid, first three rows
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
||||||
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
||||||
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
The first three rows of the first grid consists of three 3x3 sub-squares. Construct row 4 up to 6 by swapping the sequence of the three columns in the three 3x3 sub-squares into 2-3-1 (instead of 1-2-3). Construct row 7 up to 9 by swapping the sequence of the three columns in the three 3x3 sub-squares into 3-1-2 (instead of 1-2-3).
First grid
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
3 |
4 |
5 |
6 |
7 |
8 |
0 |
1 |
2 |
|
6 |
7 |
8 |
0 |
1 |
2 |
3 |
4 |
5 |
|
1 |
2 |
0 |
4 |
5 |
3 |
7 |
8 |
6 |
|
4 |
5 |
3 |
7 |
8 |
6 |
1 |
2 |
0 |
|
7 |
8 |
6 |
1 |
2 |
0 |
4 |
5 |
3 |
|
2 |
0 |
1 |
5 |
3 |
4 |
8 |
6 |
7 |
|
5 |
3 |
4 |
8 |
6 |
7 |
2 |
0 |
1 |
|
8 |
6 |
7 |
2 |
0 |
1 |
5 |
3 |
4 |
The second grid is a reflection (rotated by a quarter) of the first grid. Take 1x number from first grid +1 and add 9x number from the same cell of the second grid.
1x number + 9x number = panmagic 9x9 square
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 5 | 2 | 7 | 4 | 1 | 6 | 3 | 0 | 73 | 47 | 21 | 67 | 41 | 15 | 61 | 35 | 9 | ||||
| 3 | 4 | 5 | 6 | 7 | 8 | 0 | 1 | 2 | 6 | 3 | 0 | 8 | 5 | 2 | 7 | 4 | 1 | 58 | 32 | 6 | 79 | 53 | 27 | 64 | 38 | 12 | ||||
| 6 | 7 | 8 | 0 | 1 | 2 | 3 | 4 | 5 | 7 | 4 | 1 | 6 | 3 | 0 | 8 | 5 | 2 | 70 | 44 | 18 | 55 | 29 | 3 | 76 | 50 | 24 | ||||
| 1 | 2 | 0 | 4 | 5 | 3 | 7 | 8 | 6 | 2 | 8 | 5 | 1 | 7 | 4 | 0 | 6 | 3 | 20 | 75 | 46 | 14 | 69 | 40 | 8 | 63 | 34 | ||||
| 4 | 5 | 3 | 7 | 8 | 6 | 1 | 2 | 0 | 0 | 6 | 3 | 2 | 8 | 5 | 1 | 7 | 4 | 5 | 60 | 31 | 26 | 81 | 52 | 11 | 66 | 37 | ||||
| 7 | 8 | 6 | 1 | 2 | 0 | 4 | 5 | 3 | 1 | 7 | 4 | 0 | 6 | 3 | 2 | 8 | 5 | 17 | 72 | 43 | 2 | 57 | 28 | 23 | 78 | 49 | ||||
| 2 | 0 | 1 | 5 | 3 | 4 | 8 | 6 | 7 | 5 | 2 | 8 | 4 | 1 | 7 | 3 | 0 | 6 | 48 | 19 | 74 | 42 | 13 | 68 | 36 | 7 | 62 | ||||
| 5 | 3 | 4 | 8 | 6 | 7 | 2 | 0 | 1 | 3 | 0 | 6 | 5 | 2 | 8 | 4 | 1 | 7 | 33 | 4 | 59 | 54 | 25 | 80 | 39 | 10 | 65 | ||||
| 8 | 6 | 7 | 2 | 0 | 1 | 5 | 3 | 4 | 4 | 1 | 7 | 3 | 0 | 6 | 5 | 2 | 8 | 45 | 16 | 71 | 30 | 1 | 56 | 51 | 22 | 77 |
This 9x9 magic square is panmagic and 3x3 compact (but not symmetric).
You can use this method to construct magic squares of order is odd square (= 9, 25, 47, 81, ...).
This method gives not many possibilities (try it yourself).
It is even possible to get a symmetric result.
1x number +1 + 9x number = ultra magic 9x9 square
| 0 | 2 | 1 | 6 | 8 | 7 | 3 | 5 | 4 | 4 | 7 | 1 | 5 | 8 | 2 | 3 | 6 | 0 | 37 | 66 | 11 | 52 | 81 | 26 | 31 | 60 | 5 | ||
| 6 | 8 | 7 | 3 | 5 | 4 | 0 | 2 | 1 | 3 | 6 | 0 | 4 | 7 | 1 | 5 | 8 | 2 | 34 | 63 | 8 | 40 | 69 | 14 | 46 | 75 | 20 | ||
| 3 | 5 | 4 | 0 | 2 | 1 | 6 | 8 | 7 | 5 | 8 | 2 | 3 | 6 | 0 | 4 | 7 | 1 | 49 | 78 | 23 | 28 | 57 | 2 | 43 | 72 | 17 | ||
| 2 | 1 | 0 | 8 | 7 | 6 | 5 | 4 | 3 | 1 | 4 | 7 | 2 | 5 | 8 | 0 | 3 | 6 | 12 | 38 | 64 | 27 | 53 | 79 | 6 | 32 | 58 | ||
| 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 0 | 3 | 6 | 1 | 4 | 7 | 2 | 5 | 8 | 9 | 35 | 61 | 15 | 41 | 67 | 21 | 47 | 73 | ||
| 5 | 4 | 3 | 2 | 1 | 0 | 8 | 7 | 6 | 2 | 5 | 8 | 0 | 3 | 6 | 1 | 4 | 7 | 24 | 50 | 76 | 3 | 29 | 55 | 18 | 44 | 70 | ||
| 1 | 0 | 2 | 7 | 6 | 8 | 4 | 3 | 5 | 7 | 1 | 4 | 8 | 2 | 5 | 6 | 0 | 3 | 65 | 10 | 39 | 80 | 25 | 54 | 59 | 4 | 33 | ||
| 7 | 6 | 8 | 4 | 3 | 5 | 1 | 0 | 2 | 6 | 0 | 3 | 7 | 1 | 4 | 8 | 2 | 5 | 62 | 7 | 36 | 68 | 13 | 42 | 74 | 19 | 48 | ||
| 4 | 3 | 5 | 1 | 0 | 2 | 7 | 6 | 8 | 8 | 2 | 5 | 6 | 0 | 3 | 7 | 1 | 4 | 77 | 22 | 51 | 56 | 1 | 30 | 71 | 16 | 45 |
This 9x9 magic square is panmagic, 3x3 compact and symmetric, so it is ultra magic.
