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

I call this principle the Odd-Sided Squares Principle, developed by me. I sincerely have never seen anyone write about this, nor has anyone ever questioned the fact of me being the developer of it. I'll try to sum it up in the next lines.

The principle's rules are:
1) Only odd-sided squares are valid; in other words, squares of dimensions 1x1, 3x3, 5x5, etc.
The shape of the 3x3 square would look like this:
___ ___ ___
|_A_|_B_|_C_|
|_D_|_E_|_F_|
|_G_|_H_|_I_|

Thus, 3 rows by 3 columns; in other words, 9 cells.

2) The squares can be filled up with numbers from 1 to n² (n = dimension of square). No number can be repeated and, consequently, all numbers must be used.

3) The sum of the values of each column (in the example, A+D+G) must be the same as the sum of each other column individually. This sum must also be the same as of each row individually and each of the diagonals (A+E+I and C+E+G) individually.

These are the three rules of this principle. The 3x3 (n=3) square would look like this:
___ ___ ___
|_2_|_7_|_6_|
|_9_|_5_|_1_|
|_4_|_3_|_8_|

Each row, column and diagonal sums up to 15.

I have developed the algorithm for this, for any n-sided square (n x n), with n being an odd integer. If anyone has seen this before, please tell me, because I want credit for this principle only if I really deserve it.

Thank you,
Bruno.

Posted by Bruno on 2005-08-30 11:49:37