MS6. MAGIC SQUARES

Ralph Strachey's Method.

Mr. Ralph Strachey has devised rules for the construction of a magic square of a singly-even order n where n = 2(2m+1) , The rule states:- Divide the square into 4 equal quarters A, B, C,& D.
A B
D C
Construct in A, by the Hindu Rule a magic square with the numbers from 1 to u^2, where u = n/2. Construct by the identical method squares B, C & D using numbers from u^2+1 to 2u^2; 2u^2+1 to 3u^2; & 3u^2+1 to 4u^2(n^2). (Best done by adding u^2 to numbers in A to get B, then adding u^2 to B to get C and then adding u^2 to C to get D). Clearly the resulting composite square is magic in columns. Now in the middle row of A take the m cells next but one to the left-hand side and in each of the other rows take the m cells nearest to the left-hand side; interchange the numbers in these cells with the corresponding numbers in cells in D. Next interchange the numbers in the cells in each of the m+2 columns next to the left-hand side of B with the numbers in the corresponding cells of C. Now the resulting square will be truly a magic square. For ex. for a 6x6 square, n = 6, m = 1, u =3.
The preliminary, and the final square are as under:-
08 01 06 17 10 15 35 01 06 26 19 24
03 05 07 12 14 16 03 32 07 21 23 25
04 09 02 13 18 11 31 09 02 22 27 20
35 28 33 26 19 24 08 28 33 17 10 15
30 32 34 21 23 25 30 05 34 12 14 16
31 36 29 22 27 20 04 36 29 13 18 11
Although Mr. Strachey talks of construction of the odd order square by the Hindu rule it is not essential, provided we add u^2 to numbers in A etc., we can use any odd order square to begin with. The Hindu Rule, of course, enables us to get the odd order square quickly. A 10x10 square, using a 5x5 square constructed differently, is generated below:-
First the 10x10 order square before application of the Strachey rule:
03 016 09 22 15 28 41 34 47 40
20 008 21 14 02 45 33 46 39 27
07 025 13 01 19 32 50 38 26 44
24 012 05 18 06 49 37 30 43 31
11 004 17 10 23 36 29 42 35 48
78 091 84 97 90 53 66 59 72 65
95 083 96 89 77 70 58 71 64 52
82 100 88 76 94 57 75 63 51 69
99 087 80 93 81 74 62 55 68 56
86 079 92 85 98 61 54 67 60 73
and now the final square.
78 091 09 22 15 53 66 59 72 40
95 083 21 14 02 70 58 71 64 27
07 100 88 01 19 57 75 63 51 44
99 087 05 18 06 74 62 55 68 31
86 079 17 10 23 61 54 67 60 48
03 016 84 97 90 28 41 34 47 65
20 008 96 89 77 45 33 46 39 52
82 025 13 76 94 32 50 38 26 69
24 012 80 93 81 49 37 30 43 56
11 004 92 85 98 36 29 42 35 73