Surfaces, polyhedra and the Euler characteristic

A convex polyhedron has $30$ edges and $12$ faces. How many vertices does it have?

A football is stitched from $12$ pentagons and $20$ hexagons, with three faces meeting at every vertex. How many vertices does it have?

A plane graph consists of three separate triangles, drawn side by side without touching. Compute $V - E + F$ (remember the outer face).

Glue the sides of an octagon according to the word $a\,b\,c\,d\,a^{-1}b^{-1}c^{-1}d^{-1}$. All eight corners turn out to be glued to a single vertex. The surface is orientable. What is its genus $g$?

Which of these closed surfaces are orientable? (Choose all that apply.)

In the proof that there are only five Platonic solids we reach $(p - 2)(q - 2) < 4$. Why does this force $p \le 5$?

In the Lean theorem platonic_solids, which hypothesis encodes every face is a $p$-gon (with $F$ faces and $E$ edges)?

The hexagon word $a\,b\,a^{-1}b^{-1}c\,c$ describes the connected sum of a torus and a projective plane. By the classification theorem, which surface is it?

Write euler_char(word) for a polygon word such as 'abAB' (lower case: side traversed along its arrow; upper case: against it; each letter used twice). Return $\chi = V - E + 1$, where $V$ is the number of corner classes after gluing. Corner $i$ is the start of side $i$; a side running along its arrow goes from corner $i$ to corner $i + 1$ (mod the number of sides), one running against it goes the other way, and gluing two sides identifies their tails and their heads.