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.

  • 12345678
    910111213141516
    1718192021222324
    2526272829303132
    3334353637383940
    4142434445464748
    4950515253545556
    5758596061626364
  • 12345678
    910111213141516
    2423222120191817
    3231302928272625
    4039383736353433
    4847464544434241
    4950515253545556
    5758596061626364

In a second step, also the middle n/2 columns are reversed and a magic square is created.

125960616278
910515253541516
2423464544431817
3231383736352625
4039302928273433
4847222120194241
4950111213145556
575834566364

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.

  • 12345678
    161514131211109
    2423222120191817
    2526272829303132
    3334353637383940
    4847464544434241
    5655545352515049
    5758596061626364
  • 158360616638
    165514535211509
    2447224544194217
    2534273637303932
    3326352829383140
    4823462120431841
    5615541312511049
    572594562764

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 choose for 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.

  • 6463626160595857
    5655545352515049
    1718192021222324
    2526272829303132
    3334353637383940
    4142434445464748
    161514131211109
    87654321
  • 642624559757
    5610541213511549
    1747194544224224
    2539273736303432
    3331352928382640
    4123432120461848
    165014525311559
    858660613631

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.

  • 87654321
    910111213141516
    1718192021222324
    3231302928272625
    4039383736353433
    4142434445464748
    4950515253545556
    6463626160595857
  • 575865436364
    5655111213145049
    4847192021224241
    3334302928273940
    2526383736353132
    2423434445461817
    161551525354109
    126261605978