# 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.

•  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.

•  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.

•  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.

•  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