Intuition about the class equation (and flowers).
Solution 1:
Apparently everything is correct except the naming of the cosets.
If $cxc^{-1} \in cN(x)$ then $xc^{-1} \in N(x)$ and $c^{-1} \in N(x)$. This is a contradiction since we have to have $c \notin N(x)$ to get a non-trivial coset.
The cosets were supposed to be labelled differently. A correct labelling that agrees with the diagrams can be given by $cxc^{-1}N(x)$ (instead of $cN(x)$).