Strachey

Ralph Strachey's method divides a square of order n=4k+2 into four quadrants with odd order m=n/2, each of which is assigned a specific base number.

A C
D B
0 2 m2
3 m2 m2

The quadrant A is filled with numbers 1 to m2. Strachey chose the method of de la Loubère's, when introducing his method, but you can also choose any other method for odd order magic squares. The three other quadrants are then filled with the same magic square, where the base number belonging to the quadrant is added to each number.

Methode von Strachey (1)

This special arrangement ensures that the columns already have the same sum. However, this does not yet apply to the rows and the diagonals. To achieve this goal, a few more cells have to be replaced. It is

n=2 · m=2 · (2 · k + 1)=4k + 2

where k specifies the number of numbers per row in quadrant A, which must be exchanged with the corresponding vertically symmetrical numbers of quadrant D. The numbers are selected one after the other from the left edge of the rows. As the only exception, the first numbers in the center row of quadrants A and C remain unaffected and the k numbers from the second position are exchanged.

In quadrants B and C, on the other hand, all numbers in the k − 1 columns seen from the right edge are exchanged with the vertically symmetrical number from the lower quadrant. In the case of 6th-order is k − 1=0, so that these exchanges are only used from order n=10.

Since only numbers in one column are swapped, the column sums do not change. But the rows and diagonal sums are now also magic, since the sum of the basic numbers in the rows and diagonals is now always m2 after the swaps. Added are the numbers from the square of de la Loubère, so that the resulting square is magic.

  • 816261924
    357212325
    492222720
    352833171015
    303234121416
    313629131811
  • 3516261924
    3327212325
    3192222720
    82833171015
    30534121416
    43629131811

In the example shown for n=6 there were no exchanges on the right edge. So another example for order n=10 shall be given. This time the method of Moschopoulos is chosen as the base square for quadrant A.

Methode von Strachey (3)

This is followed by swaps on the left and right edges to balance the base numbers in rows and diagonals. To do this, k numbers on the left margin and k − 1 numbers on the right margin must be exchanged vertically symmetrically

  • 112472036174577053
    412258165462755866
    175132196755637159
    1018114226068516472
    236192157356695265
    86998295783649324528
    798710083912937503341
    92808896844230384634
    85937689973543263947
    98819477904831442740
  • 869972036174577028
    7987258165462755841
    1780882196755637134
    8593114226068516447
    9881192157356695240
    11248295783649324553
    41210083912937503366
    9251396844230384659
    10187689973543263972
    2369477904831442765

and the magic square is created.