Mathematical Olympiad Treasures Problem 1.11 Let $n$ be a positive integer, prove that:
$3^{3^{n}}(3^{3^{n}}+1) + 3^{3^{n} + 1} - 1$ 
is not prime
The solution states:
Observe:
$3^{3^{n}}(3^{3^{n}}+1) + 3^{3^{n} + 1} - 1 = a^{3} + b^{3} + c^{3} - 3abc$
for $a = 3^{3^{n-1}}$, $b = 9^{3^{n-1}}$, $c = -1$
After that main insight the solution follows quite simply.
How are you supposed to find what $a$ and $b$ are. It makes sense to guess that $c = -1$ but finding $a$ and $b$ seems much harder, how do you guess it?

$$3^{3^{n}}(3^{3^{n}}+1) + 3^{3^{n} + 1} - 1=\\
3^{2\cdot3^n}+3^{3^n}-1+3^{3^n+1}=\\
9^{3^n}+3^{3^n}-1+3^{3^n+1}\\
(9^{3^{n-1}})^3+(3^{3^{n-1}})^3+(-1)^3-3(9^{3^{n-1}}\cdot 3^{3^{n-1}}\cdot(-1))$$
