Can you partition the numbers 1, 2, 3, ... nē in n separate subsets, each with n numbers, all subsets having the same sum?

For even n, arrange the numbers 1 to n^2 in an array as follows.  Go from left to right the first row, right to left the second row, left to right the third row, etc.

1       2      3         . . . n
2n     2n-1  2n-2    . . . n+1
2n+1 2n+2 2n+3   . . . 3n
4n     4n-1   4n-2   . . . 3n+1
.
.
.
n^2   n^2-1 n^2-2 . . . n(n-1)+1

Column sums are all the same because numbers go up in odd rows and down in even rows by the same amount in a given column.

Of course, this method solves partitioning 1 to nk in k seperate subsets, each with n numbers, all subsets having the same sum when n is even.

Posted by McWorter on 2005-08-31 18:27:44