The three cottage problem is a problem in mathematical graph theory:

Suppose there are three cottages that each need to be connected to the gas, water, and electric companies. Is there a way to do so without any of the lines crossing each other?

In more formal terms, this asks whether the bipartite graph K_3,3 is planar. Kazimierz Kuratowski proved in 1930 that K_3,3 is nonplanar, and thus that the three cottage problem has no solution.