Solution :

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