  We show a few nontrivial extensions in the classical Adams spectral sequence.
In particular, we compute that the 2-primary part of $\pi_{51}$ is
$\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$. This was the last unsolved
2-extension problem left by the recent works of Isaksen and the authors
(\cite{Isa1}, \cite{IX}, \cite{WX1}) through the 61-stem.
  The proof of this result uses the $RP^\infty$ technique, which was introduced
by the authors in \cite{WX1} to prove $\pi_{61}=0$. This paper advertises this
method through examples that have simpler proofs than in \cite{WX1}.
