MS8. MAGIC SQUARES

De La Hire's Method:

De La Hire has kmnnsuggested a modified form of this procedure for even order square of 6_6. His method is to form two auxiliary squares with 0,6,12,18,24 & 30; and 1,2,3,4,5 & 6 in such a way that the columns of one square and the rows of the other square each so contain numbers, 3 times repeated, that the required sum summation of 90 for one swquare and 21 for the other square is achieved. We may, however, form these squaeres using A's and a's using the conditions stipulated by him. Our condition then will be that A's and a's be so combined in pairs that they total 30 and 7 respectively. We can take combinations of 0 and 30; 6 and 24; 12 and 18; 1 and 6; 2 and 5; 3 and 4. This will enable us to use permutations of numbers and thus generate multiple squares. The auxiliary squaress will take this shape:
A F F A F A ............. a e d c b f
E B E E B B .............. f b d c e a
D C C C D D ............ f e c d b a
C D D D C C ............ a e c d b f
B E B B E E .............. f b c d e a
F A A F A F ............. a b d c e f
If we take A = 0, B = 6, C = 12, D =18, E = 24, F = 30, a =1, b = 2, c = 3, d = 4, e = 5 and f = 6, our 6x6 square will be:
01 35 34 03 32 06
30 08 28 27 11 07
24 17 15 16 20 19
13 23 21 22 14 18
12 26 09 10 29 25
31 02 04 33 05 36
A further modification of this method, a bit more complicated, permits us use of A's and a's in such a way that the arithmetical sum of each row, column, and each diagonal, with appropriate values of A's and a's total nxn(n-1)/2 for A's and nx(n+1)/2 for a's. This means that there can be 2 or more A's and a's in diagonals too.