Sum of three perfect cubes is equal to a perfect fourth How many answers does the following equation have?
$a^3+b^3+c^3=d^4$ where $a, b, c, d\in \mathbb{Z}^+$.
This was asked in a test for gifted math students in 7th grade in Finland. I have been thinking about this and couldn't solve this for my number theory isn't that good.
Could someone help me? Thank you in advance.

It has an infinite number of solutions. For instance, for any $k\in \mathbb{N}$, take $d=3k^3$, and $a=b=c=3k^4$. 
Then
$$ a^3+b^3+c^3=3\cdot 3^3k^{12} = 3^4k^{12}=d^4.$$
