There is no planar solution: the topology of the plane forbids it. There's been proofs of this for more than 60 years (such as in the paper: G. T. Kneebone, "The Three Houses Problem"The Mathematical Gazette, Vol. 25, No. 264 (May, 1941), pp. 78-81). It can, however, be done on a surface topologically equivalent to the torus, or other multiply-connected surfaces.

--Kevin C.