So, I know that $T^2 \# RP^2 = RP^2 \# RP^2 \# RP^2$, and I've seen a proof of this which I felt like I understood (rearranging fundamental polygons). But trying to come up with it myself I ended up with this "proof" that $T^2 \# RP^2 = RP^2 \# RP^2$.
My thought process was: write $T^2 \# RP^2$ as being represented by the polygon $aba^{-1}b^{-1}cc$, cut along the horizontal which starts at the vertex between the two "$c$" edges and ends at the vertex between $b$ and $a^{-1}$, flip the bottom half-hexagon, and attach along $c$, yielding $abd^{-1}bad$, which we can pull $aa$ out of to get $T^2 \# RP^2 = RP^2 \# S$, where $S$ is represented by $bd^{-1}bd^{-1}$, which we can pull $bb$ out of to get $S = RP^2 \# S'$, where $S'$ is represented by $dd^{-1}$, and so $S = RP^2$ and $T^2 \# RP^2 = RP^2 \# RP^2$. Where did I go wrong?