No, you are misinterpreting the problem.

You have 36 ordered pairs (regiment, rank), where you might label the regiments from 1 to 6 and the ranks from 1 to 6, so the pairs are of the form (i,j), i = 1 to 6, j = 1 to 6. You have to arrange the pairs in a 6 by 6 array so that if you consider the regiments alone, each regiment appears exactly once in each row and column, and if you consider the ranks alone, each rank appears exactly once in each row and column.

Here is a solution for the 3 by 3 case:

Code:

(1,1) (2,2) (3,3)
(3,2) (1,3) (2,1)
(2,3) (3,1) (1,2)

As you may know, the 6 by 6 case has been shown to be impossible.