> 
> Let $x\in\mathbb{R}$. Say $x$ is satanic if $x$ contains "$666$" somewhere in its decimal expansions. Let $S$ be the set of such numbers. Is
> $$
> \mathbb{R} \setminus \mathbb{Q} \subseteq S 
> $$
> More generally, for any finite string of digits, say $y = y\_1y\_2y\_3\ldots y\_{n-1}y\_n$, does $x\in \mathbb{R}\setminus\mathbb{Q}$ necessarily contain $y$ in its decimal expansion.
> 
> 
> 

It seems intuitive that, at least, for any $y$, $y$ should be contained in $\textit{almost every }$ irrational number, since the irrational numbers have an infinite decimal expansion. But, I could envision there exists some irrational number that follows some pattern such that it doesn't contain some $y$.