Since the riddle talks about one-twelfth of Diophantus' life and one-seventh of his life, we guess that the number of years he lived is a multiple of both $12$ and $7$, and hence a multiple of $84$. But the only multiple of $84$ reasonable for a human lifespan is $84$, and we see that $84$ years satisfies the conditions of the problem: $\frac{84}{6}+\frac{84}{12}+\frac{84}{7}+5+\frac{84}{2}+4=14+7+12+5+42+4=84.$ The reasonable guess succeeds. We can also solve the problem with algebra. Let $x$ be the number of years Diophantus lived. Then $\frac{x}{6}+\frac{x}{12}+\frac{x}{7}+5+\frac{x}{2}+4=x.$ Hence $9=x\left(\frac{9}{84}\right),$ and $x=84$.