Strachey method for magic squares
# Strachey method for magic squares

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:

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 n = 4k + 2.

1. Divide the grid into 4 quarters each having $n^2/4$ 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 2k + 1 in subsquares A, B, C, D, first filling up the sub-square A with the numbers 1 to $n^2/4,$ then the sub-square B with the numbers $n^2/4$ + 1 to $2n^2/4,then$ the sub-square C with the numbers $2n^2/4$ + 1 to $3n^2/4,$ then the sub-square D with the numbers $3n^2/4$ + 1 to $n^2.$ 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 | 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 k 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 k - 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 n=4k + 2.

