Nails and Rubber Bands. That is the suggestive name of a game played by a group of children (all of them offspring of geometry teachers). The children fix a number of nails on a plank of wood, randomly placed. Then they choose one of the nails to be the Origin, and a number B of rubber bands.
The challenge is to use the B rubber bands to wrap the nails so that (i) each rubber band wraps a subset of the nails; (ii) all nails are inside some wrapping; (iii) wrappings do not overlap each other except at the Origin nail, which is touched by all rubber bands; (iv) rubber bands must form wrappings which are convex polygons with at least three corners; and (v) the total area inside the wrappings is the smallest among all possible ways of wrapping the nails. An instance of the game is shown in Figure 1.