Strachey method for magic squares

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

Strachy'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 subsquare A with the numbers 1 to k^2/4, then the subsquare B with the numbers k^2/4 +1 to 2*k^2/4,then the subsquare C with the numbers 2*k^2/4 +1 to 3*k^2/4, then the subsquare 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 subsquare A with the corresponding columns of subsquare D 92 "99 1 8 15 67 74 51 58 65

98 807 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 subsquare C with the corresponding columns of subsquare 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 subsquare A with the corresponding cell of subsquare D. Exchange the central cell in subsquare A with the corresponding cell of subsquare 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

From W W Rouse Ball Mathematical Recreations and Essays, (1911)

ee also

*Conway's LUX method for magic squares
*Siamese method