Problem: are two points in the same polygon?
Solution: a line drawn between those two points has an odd number of intersections with the polygon if they are both in it (or outside of it), and even number if they are not.
That's under-informative, we only know that both points are in the same polygon or outside of it. The underlying elegant algorithm is that of knowing whether a point is inside a polygon: a ray from that point out to infinity intersects the polygon an odd number of times if the point is in it, even otherwise.