How to prove that the problem of the three houses on the plane has no solution. We should add three more paths to the hexagon that join every vertex with its opposite. One of these paths will go through the region and block the way for the other two; the second one will have to pass outside the region and block the way for the other two as well. The third paths can't cross either the interior and the exterior of the region.
You can also try to solve this problem online with the interactive animation Paths without crossings.
The image belongs to the sections...:
From "fragments of topology" (From "XlaTangente")
Jordan's theorem and the three houses problem (Topology)
Images from the topology section (From the exhibitions of matematita)