As we know that $2(a^2+b^2)\geq (a+b)^2$ or in our problem $2n^3\geq n^4$ or $2\geq n$ so n can be n=0,1 or 2

One still has to exhibit the representations. We have $0^2 = 0+0$ and $0^3 = 0^2+0^2$; then $1^2 = 0+1$ and $1^3 = 0^2+1^2$; lastly $2^2 = 2+2$ and $2^3 = 2^2+2^2$.

Is this from a 2014 TST? This problem is from 2004 Romanian MO.

True Moreover, Problem 1 - $8$-th grade.