Two aunts are living each in her own (0-dimensional) house. There are two non-intersecting (1-dimensional) roads between the houses.

Last year, both aunts were doing a lot of exercise, and so they were slim (0-dimensional). They managed to walk together from House 1 to House 2, taking different roads, while each was holding one end of a rope of length less than L.

This year, they ate a ton of doughnuts and gained weight, so each became a sphere of radius \(\frac{L}{2}\). One aunt is in House 1 and the other is in House 2. Can they exchange houses without bumping into each other (their centers must always remain on the roads)?