= Solution
Use the same $4\times4$ square grid with four centre-based triangles per small square, but identify boundary points by
$$
(x,y)\sim(1-x,1-y)\qquad\text{for }(x,y)\in\partial Q.
$$
This gives the <real projective plane>. To see the model explicitly, a radial <homeomorphism> from the square, centred at $(1/2,1/2)$, to the <closed disc> carries the boundary involution to the <antipodal map>. The map from that disc to the <projective plane>
$$
(u,v)\longmapsto [u:v:\sqrt{1-u^2-v^2}]
$$
is <continuous> and <surjective>: every one-dimensional <vector subspace> of $\mathbb R^3$ has a <unit vector> representative in the closed upper hemisphere. Its only identifications are antipodal boundary points. The <compact-to-Hausdorff continuous bijection theorem> therefore identifies this disc <quotient space> with $\mathbb{RP}^2$.
The square <triangulation> descends to an abstract <simplicial complex> $K_2$. Small-square centres remain distinct interior vertices; each triangle has one centre and two adjacent grid vertices. Boundary pairings identify whole grid edges, while preserving distinct vertices in each triangle, so this is a finite <triangulation> of the <real projective plane>. Its counts are $33$ vertices, $96$ edges and $64$ triangles, giving <Euler characteristic> one.
Choose $\sigma$ to be the interior triangle with vertices $(1/4,1/4)$, $(1/2,1/4)$ and $(3/8,3/8)$. Removing its <relative interior> and cutting from $(0,0)$ to $(1/4,1/4)$ gives the same nine-sided cut disc as before. Parametrize it by a <convex polygon>, transporting the boundary identifications. Its boundary word is now
$$
\boxed{a\,b\,a\,b\,c\,d\,e\,f\,c^{-1}}.
$$
The boundary involution pairs the two $a$ sides and the two $b$ sides in matching traversal directions; the $c$ pair restores the slit. Leave $d,e,f$ unpaired. The <quotient space> is
$$
Y=|K_2|\setminus\operatorname{inside}(\sigma),
$$
the <real projective plane> with an open disc removed. In particular it is a <compact> <connected> <surface with boundary> consisting of one three-edge <circle>.
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