I was watching a Numberphile video and I saw how Mathematicians were able to express almost any number as the sum of three cubes$$\begin{align*}1 & =1^3+0^3+0^3\\ & =9^3+10^3+(-12)^3\\29 & =3^3+1^3+1^3\\53 & =27^3+27^3+(-1)^3\\51 & =659^3+602^3+(-796)^3\end{align*}$$However, I'm wondering how do you generate these types of values. It's obvious for small-valued numbers, but some can get really exhausting$$30=(2\,220\,433\,932)^3+(-2\,218\,888\,517)^3+(-283\,069\,966)^3$$
Question: How are these kinds of solutions generated?