# Strachey   (Variant of Benson and Jacoby)

Benson and Jacoby have modified Strachey's method so that many different magic squares of order n=2m=4k + 2 can be generated. The entire square is divided into four quadrants again.

 A C D B
 0 2 m2 3 m2 m2

These quadrants are also filled with the basic numbers 0 , m2 , 2 m2und 3 m2 as before. However, the individual quadrants are no longer considered separately, but the entire upper half with quadrants A and B. To ensure that the basic numbers are equalized, there are two options for the number of occurrences of these numbers in rows, which are listed in the table.

 0 m2 2 m2 3 m2 entweder k k + 2 k − 1 k + 1 oder k + 1 k − 1 k + 2 k

In both cases, row sums are the same. However, two other conditions have to be met in order to create a magic square from such an arrangement of base numbers.

• In each row, the base numbers, which are exactly n/2 columns apart, must not be identical or complementary to 3 m2. For example, if with order n=10 the number 25 exists in column 1, column 6 must have either the number 0 or 75.
• The sum of the numbers on the secondary diagonal of quadrant A must be equal to the sum of the numbers on the main diagonal of B.

If all the numbers in the upper half are chosen in accordance with these conditions, they are mirrored on the horizontal center row, where they are replaced by the numbers which are complementary with respect to 3 m2.

•  0 0 50 75 50 25 50 75 50 0 50 75 50 50 75 0 25 0 0 50 0 75 25 0 75 25 25 75 25 50 50 50 50 0 75 75 0 0 25 50 0 75 50 25 0 25 25 75 75 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
•  0 0 50 75 50 25 50 75 50 0 50 75 50 50 75 0 25 0 0 50 0 75 25 0 75 25 25 75 25 50 50 50 50 0 75 75 0 0 25 50 0 75 50 25 0 25 25 75 75 25 75 0 25 50 75 50 50 0 0 50 25 25 25 75 0 0 75 75 50 25 75 0 50 75 0 50 50 0 50 25 25 0 25 25 0 75 50 75 75 25 75 75 25 0 25 50 25 0 25 75

Finally, only the numbers of a existing magic square of odd order have to be added. In the top two quadrants you take any magic square, while before inserting it in the bottom two quadrants, it has to be mirrored vertically. For example, if you select the Moschopoulos square of order m=5, these changes from Benson and Jacoby will create the magic square in the figure below.

•  11 24 7 20 3 4 12 25 8 16 17 5 13 21 9 10 18 1 14 22 23 6 19 2 15
•  11 24 57 95 53 36 74 82 70 3 54 87 75 58 91 4 37 25 8 66 17 80 38 21 84 42 30 88 46 59 60 68 51 14 97 85 18 1 39 72 23 81 69 27 15 48 31 94 77 40 98 6 44 52 90 73 56 19 2 65 35 43 26 89 22 10 93 76 64 47 92 5 63 96 9 67 55 13 71 34 29 12 50 33 16 79 62 100 83 41 86 99 32 20 28 61 49 7 45 78