At Quiz Master Shop Towers we have a square lawn surrounded on all four sides by the building. At each corner of the lawn there is a door in the wall, each leading to different wings of our commodious abode. Each side of the lawn is ten metres.
We are fed up with visitors wandering hither and thither across the lawn when going from door to door, and want to put down paths to reduce the wear on the grass.
The thing is, upkeep on these large country piles can be expensive, so we want to use as little paving as we can get away with. We want to put down the shortest total length of paths possible, while providing a route between every pair of doors.
Putting a path around the outside would provide the routes, but that's 40 metres of paving. Leaving out one side gets it down to 30 metres, but some recalcitrants would not walk all the way round when going between the pair of doors without a direct path. And this isn't the shortest solution.
The solution isn't two diagonals either which total just over 28 metres (20 times the square root of 2).
The best solution is two paths at 30 degrees to one of the sides that meet, and then a path from there to the centre. With this mirrored in the other half of the square you end up with a "bowtie shape". The path length is 27.32 metres.
This is part of two adjoining hexagons!