Tethering in pastures new

Extract

In a recent and illuminating article that provided much food for thought [1], the problem of tethering a goat at the edge of a circular pasture so as to restrict its attentions to only one half of the grazing supply was elegantly addressed and developed further to embrace the corresponding three-dimensional scenario involving a bird. The exercises resulted in mathematical formulations that required the use of numerical methods to extract practical results. Following the article, various questions and different scenarios sprang to my mind. The following poser perhaps best illustrates one of these, and it is the purpose of this Article to address this particular conundrum:

A grazier has three troublesome beasts that are water averse, eat grass and who will, given half a chance, eat one another also in some fashion. The first will eat the other two and the second will eat only the third, which eats just grass. Having stabled and fed them in separate stalls during the winter months he plans to release them in the spring to an arbitrarily elliptic shaped pasture up to the water's edge in the middle of a lake. He has at his disposal:

1. (1) A drum of tethering rope from which he can cut just once any required length, TBD (To Be Determined).

2. (2) Slip rings and two tethering pegs that can be positioned only on the pasture boundary (i.e. at the water's edge).