Can the boundaries of two pentagons intersect at $20$ points?

Solution 1:

According to Theorem 4 from Černy et al.'s "On the number of intersections of two polygons" (2003), the number of intersections is bounded by $$5\times 5-\left\lceil \frac56 \right\rceil-5=19$$ Therefore $18$ is the maximum.

Alternatively, Theorem 5 gives the exact value $4\times 5-2=18$.

These maximal intersection numbers are hard to compute when both polygons have an odd number of vertices. Otherwise there are general formulas given by Theorem 1 and 2.

Edit: found a better result in Günther's "The maximum number of intersections of two polygons" (2012).

Theorem: the maximal number of intersections between an $n$-polygon and an $m$-polygon with $n$ and $m$ both odd is $(n-1)(m-1)+2$.