Since  ,  is divisible by 3 and therefore n is congruent to one of 0, 2, 3, 5 modulo 6.

Now we prove that if n is congruent to one of 0, 2, 3, 5 modulo 6  and n > 4 then such partition exists.

If we can find such partition for some n then we can enlarge it to an admissible partition for n+6 by adjoining n+1 , n+6 to A; n+2, n+5 to B; n+3, n+4 to C. For n = 5, 6, 8, 9 we have the following partitions

n = 5                A = {1,4}                    B = {2,3}         C = {5}

n = 6                A = {1,6}                    B = {2,5}         C = {3,4}

n = 8                A = {1,2,3,4}              B = {5,7}         C = {4,8}

n = 9                A = {1,2,3,4,5}           B = {7,8}         C = {6,9}

Obviously, for  such a partition does not exists