Graph Coloring · Challenge problem

The $5$-cycle $1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}1$ has $\chi(C_5)=3$. Among the $5$ possible "chords" (edges joining two non-adjacent rim vertices: $13,14,24,25,35$), for how many does ADDING that single chord to $C_5$ raise the chromatic number above $3$?