In a field, there are two pegs, \(A\) and \(B\), placed \(15\) metres apart. Each peg has a small ring on top, and a rope can slide freely through these rings. You have one rope and two goats that want to graze the grass, but they will fight each other if they can both reach the same spot.
The rope may be arranged in any way you choose: it could pass through both rings, only one, or neither, depending on your setup. For what lengths of rope can you arrange things so that the goats cannot fight each other?