Planck: Method of reversions
With this method, you can easily create magic squares of the order n=4k. This method by C. Planck was first introduced in the book by W.S. Andrews. You have to start with a square in a natural order, where the n/2 rows in the center of the square are reversed.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 -
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 24 23 22 21 20 19 18 17 32 31 30 29 28 27 26 25 40 39 38 37 36 35 34 33 48 47 46 45 44 43 42 41 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
In a second step, also the middle n/2 columns are reversed, and a magic square is created.
| 1 | 2 | 59 | 60 | 61 | 62 | 7 | 8 |
| 9 | 10 | 51 | 52 | 53 | 54 | 15 | 16 |
| 24 | 23 | 46 | 45 | 44 | 43 | 18 | 17 |
| 32 | 31 | 38 | 37 | 36 | 35 | 26 | 25 |
| 40 | 39 | 30 | 29 | 28 | 27 | 34 | 33 |
| 48 | 47 | 22 | 21 | 20 | 19 | 42 | 41 |
| 49 | 50 | 11 | 12 | 13 | 14 | 55 | 56 |
| 57 | 58 | 3 | 4 | 5 | 6 | 63 | 64 |
Variant 1
With this construction, the middle n/2 rows and columns do not necessarily have to be reversed. You can as well choose any n/4 rows from the lower half of the square and reverse them and their vertically symmetrical partner rows. Any n/4 columns from the left half of the square and their horizontally symmetrical partner columns must also be reversed. The result is always a magic square.
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1 2 3 4 5 6 7 8 16 15 14 13 12 11 10 9 24 23 22 21 20 19 18 17 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 48 47 46 45 44 43 42 41 56 55 54 53 52 51 50 49 57 58 59 60 61 62 63 64 -
1 58 3 60 61 6 63 8 16 55 14 53 52 11 50 9 24 47 22 45 44 19 42 17 25 34 27 36 37 30 39 32 33 26 35 28 29 38 31 40 48 23 46 21 20 43 18 41 56 15 54 13 12 51 10 49 57 2 59 4 5 62 7 64
Variant 2
As with the first variant, any n/4 rows from the lower half of the square and their vertically symmetrical partner rows are selected. These n/2 rows are now rotated by 180°. Then you select n/4 columns from the left half as well as their horizontally symmetrical partner columns and also rotate them by 180°. Then another magic square was created.
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64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 -
64 2 62 4 5 59 7 57 56 10 54 12 13 51 15 49 17 47 19 45 44 22 42 24 25 39 27 37 36 30 34 32 33 31 35 29 28 38 26 40 41 23 43 21 20 46 18 48 16 50 14 52 53 11 55 9 8 58 6 60 61 3 63 1
Variant 3
In this variant, the rows and columns are selected as in the other two variants, but processed differently. A magic square is created when you reverse the rows and rotate the columns by 180° or vice versa. In the example shown, the selected rows were reversed and the columns rotated.
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8 7 6 5 4 3 2 1 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 32 31 30 29 28 27 26 25 40 39 38 37 36 35 34 33 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 64 63 62 61 60 59 58 57 -
57 58 6 5 4 3 63 64 56 55 11 12 13 14 50 49 48 47 19 20 21 22 42 41 33 34 30 29 28 27 39 40 25 26 38 37 36 35 31 32 24 23 43 44 45 46 18 17 16 15 51 52 53 54 10 9 1 2 62 61 60 59 7 8