= Solution
Take a hexagonal <polygonal schema> with <boundary> word
$$
\boxed{a\,a\,b\,b\,c\,c.}
$$
Identify equally labelled <polygon edges> in the indicated directions. The two consecutive occurrences of $a$ identify the first three vertices; the two $b$ edges extend this class to the next two vertices, and the $c$ edges include the remaining vertex. Thus all six <polygon vertices> become one, while the six edges become three and the interior is one face. Its <Euler characteristic> is
$$
\boxed{\chi(Y)=1-3+1=-1.}
$$
This is the usual <polygonal schema> for the <connected sum of surfaces> of three <real projective planes>. The same-sign edge pairs force a <nonorientable surface>, consistently with $\chi(N_3)=2-3=-1$. A closed <orientable surface> cannot have odd <Euler characteristic>.
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