The engineer is first. She herds the sheep into a circle and then puts the fence around them, declaring, "A circle will use the least fence for a given area, so this is the best solution."
The physicist is next. He creates a circular fence of infinite radius around the sheep, and then draws the fence tight around the herd, declaring, "This will give the smallest circular fence around the herd."
The mathematician is last. After giving the problem a little thought, she puts a tiny fence around herself and declares, "I define myself to be on the outside!"