The Strachey method for magic squares is an algorithm for generating magic squares of singly even order 4k + 2. An example of magic square of order 6 constructed with the Strachey method:

Example35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28 33 17 10 15 30 5 34 12 14 16 4 36 29 13 18 11

Strachey's method of construction of singly even magic square of order n = 4k + 2.

1. Divide the grid into 4 quarters each having n2/4 cells and name them crosswise thus

A CD B

2. Using the Siamese method (De la Loubère method) complete the individual magic squares of odd order 2k + 1 in subsquares A, B, C, D, first filling up the sub-square A with the numbers 1 to n2/4, then the sub-square B with the numbers n2/4 + 1 to 2n2/4, then the sub-square C with the numbers 2n2/4 + 1 to 3n2/4, then the sub-square D with the numbers 3n2/4 + 1 to n2. As a running example, we consider a 10×10 magic square, where we have divided the square into four quarters. The quarter A contains a magic square of numbers from 1 to 25, B a magic square of numbers from 26 to 50, C a magic square of numbers from 51 to 75, and D a magic square of numbers from 76 to 100.

17 24 1 8 15 67 74 51 58 6523 5 7 14 16 73 55 57 64 664 6 13 20 22 54 56 63 70 7210 12 19 21 3 60 62 69 71 53 11 18 25 2 9 61 68 75 52 59 92 99 76 83 90 42 49 26 33 40 98 80 82 89 91 48 30 32 39 41 79 81 88 95 97 29 31 38 45 47 85 87 94 96 78 35 37 44 46 28 86 93 100 77 84 36 43 50 27 34

3. Exchange the leftmost k columns in sub-square A with the corresponding columns of sub-square D.

92 99 1 8 15 67 74 51 58 6598 80 7 14 16 73 55 57 64 6679 81 13 20 22 54 56 63 70 7285 87 19 21 3 60 62 69 71 5386 93 25 2 9 61 68 75 52 5917 24 76 83 90 42 49 26 33 4023 5 82 89 91 48 30 32 39 414 6 88 95 97 29 31 38 45 4710 12 94 96 78 35 37 44 46 2811 18 100 77 84 36 43 50 27 34

4. Exchange the rightmost k - 1 columns in sub-square C with the corresponding columns of sub-square B.

92 99 1 8 15 67 74 51 58 4098 80 7 14 16 73 55 57 64 4179 81 13 20 22 54 56 63 70 4785 87 19 21 3 60 62 69 71 2886 93 25 2 9 61 68 75 52 3417 24 76 83 90 42 49 26 33 6523 5 82 89 91 48 30 32 39 664 6 88 95 97 29 31 38 45 7210 12 94 96 78 35 37 44 46 5311 18 100 77 84 36 43 50 27 59

5. Exchange the middle cell of the leftmost column of sub-square A with the corresponding cell of sub-square D. Exchange the central cell in sub-square A with the corresponding cell of sub-square D.

92 99 1 8 15 67 74 51 58 4098 80 7 14 16 73 55 57 64 414 81 88 20 22 54 56 63 70 4785 87 19 21 3 60 62 69 71 2886 93 25 2 9 61 68 75 52 3417 24 76 83 90 42 49 26 33 6523 5 82 89 91 48 30 32 39 6679 6 13 95 97 29 31 38 45 7210 12 94 96 78 35 37 44 46 5311 18 100 77 84 36 43 50 27 59

The result is a magic square of order n=4k + 2.W W Rouse Ball Mathematical Recreations and Essays, (1911)

References

See also Conway's LUX method for magic squares Siamese method