Use all the numbers 1 to 32 to form two 4x4 magic squares with the same magic constant.

I used the fact that "joining" two orthogonal latin squares, and with a proper change of base, we arrive at a magic square.

It has no difficulty to construct two 4x4 orthogonal latin squares, that joined give us:

        00  12  23  31
        21  33  02  10
        32  20  11  03
        13  01  30  22

Observe that in each row, in each column, in each of the two diagonals, each digit 0, 1, 2, and 3, appears once as the first digit, as well as the second digit.

Now, to transform this in a magic square, we assume that the numbers are written in base 4, and change them to base 10. And since 00 in base 4 is equal to 0 in base 10, after the change of base, we must add 1 unit to every number, so that the magic square have "1" as his least number. We achieve:

       01  07  12  14
       10  16  03  05
       15  09  06  04
       08  02  13  11

This magic square uses only the numbers from 1-16, and its magic sum is 34. Adding 16 units to each number, we achieve a new magic square with the numbers 17 to 32, and the magic sum being 34 + 4*16 = 98.

       17  23  28  30
       26  32  19  21
       31  25  22  20
       24  18  29  27

Itīs easy to see now that all we have to do, since 98 - 34 = 64, is to interchange between the two magic squares, two numbers from each row, each colum, and each main diagonal.We can make this by many ways. Choosing the two outer numbers of the first and third rows, and the two inner numbers of the second and fourth rows, we finally arrive at:

        17  07  12  30      01  23  28  14
        10  32  19  05      26  16  03  21
        31  09  06  20      15  25  22  04
        08  18  29  11      24  02  13  27

Being the magic sum 66, as it should be (34 + 98)/2 or the sum from the numbers 1 to 32, divided by 8.

Edited on November 15, 2005, 3:09 pm

Posted by pcbouhid on 2005-11-15 15:06:30