Strachey Method For Magic Squares

The Strachey method for magic squares is an algorithm for generating magic squares of singly even order 4n+2.

Example of magic square of order 6 constructed with the Strachey method:

Example
35 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 k=4*n+2

1. Divide the grid into 4 quarters each having k^2/4 cells and name them crosswise thus

A C
D B

2. Using the Siamese method (De la Loubère method) complete the individual magic squares of odd order 2*n+1 in subsquares A, B, C, D, first filling up the sub-square A with the numbers 1 to k^2/4, then the sub-square B with the numbers k^2/4 +1 to 2*k^2/4,then the sub-square C with the numbers 2*k^2/4 +1 to 3*k^2/4, then the sub-square D with the numbers 3*k^2/4 +1 to k^2.

17 24 1 8 15 67 74 51 58 65
23 5 7 14 16 73 55 57 64 66
4 6 13 20 22 54 56 63 70 72
10 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 n columns in sub-square A with the corresponding columns of sub-square D.

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

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

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

The result is a magic square of order k=4*n+2.

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    He had not the method of making a fortune.”
    —Thomas Gray (1716–1771)

    “You never see animals going through the absurd and often horrible fooleries of magic and religion.... Dogs do not ritually urinate in the hope of persuading heaven to do the same and send down rain. Asses do not bray a liturgy to cloudless skies. Nor do cats attempt, by abstinence from cat’s meat, to wheedle the feline spirits into benevolence. Only man behaves with such gratuitous folly. It is the price he has to pay for being intelligent but not, as yet, quite intelligent enough.”
    —Aldous Huxley (1894–1963)

    “And New York is the most beautiful city in the world? It is not far from it. No urban night is like the night there.... Squares after squares of flame, set up and cut into the aether. Here is our poetry, for we have pulled down the stars to our will.”
    —Ezra Pound (1885–1972)