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
3516261924
3327212325
3192222720
82833171015
30534121416
43629131811

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

AC
DB

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.

172418156774515865
235714167355576466
461320225456637072
1012192136062697153
111825296168755259
92997683904249263340
98808289914830323941
79818895972931384547
85879496783537444628
869310077843643502734

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

929918156774515865
9880714167355576466
79811320225456637072
8587192136062697153
869325296168755259
17247683904249263340
2358289914830323941
468895972931384547
10129496783537444628
111810077843643502734

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

929918156774515840
9880714167355576441
79811320225456637047
8587192136062697128
869325296168755234
17247683904249263365
2358289914830323966
468895972931384572
10129496783537444653
111810077843643502759

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.

929918156774515840
9880714167355576441
4818820225456637047
8587192136062697128
869325296168755234
17247683904249263365
2358289914830323966
7961395972931384572
10129496783537444653
111810077843643502759

The result is a magic square of order n=4k + 2.[1]

References

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

See also

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