(I) Determine the total number of ways in which 2011 (base ten) is expressible as the sum of squares of N distinct positive integers, whenever N = 3, 4, 5,...., 8, 9,10

(II) Keeping all the other conditions in (I) unaltered, what are the respective total number of ways if the N positive integers need not be distinct?

The following table shows the total number of ways where
s₁² + s₂² + s₃² + ... + s_N² sums to 2011, such that
s₁ < s₂ < s₃ < ... < s_N.  (I.e., it is assumed that the problem is asking for the number of combinations of squares that sum to 2011 and not the number of permutations.)

 N    (I) 
 3 :    3
 4 :   34
 5 :  343
 6 : 1165
 7 : 2296
 8 : 3425
 9 : 4402
10 : 4739

Edited on June 27, 2011, 11:22 am

Posted by Dej Mar on 2011-06-27 11:18:20