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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
3
31
8
30
4

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

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 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.

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.

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.

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.

11	18	100	77	84	36	43	50	27	59

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

References

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

Wikipedia Source

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