Let n be the smallest positive integer such that n(n+1)(n+2)(n+3) can be expressed as either a perfect square or a perfect cube (not necessarily both).

Find n, or prove that this is not possible.

Out of any four consecutive integers, one will be a multiple of 4, and other will be even, but not multiple of 4. Thus, the product of the four integers will be even, but NOT a multiple of four -- so it cannot be a square.