How to prove that the problem of the three houses has no solution on the plane.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.
You can also try to solve this problem online with the interactive animation Paths without crossings.
The image belongs to the sections...:
Jordan's theorem and the three houses problem (Topology)
Images from the topology section (From the exhibitions of matematita)