Problem:
There is a farmer who has a $1\text{ mile}\times 1\text{ mile}$ square piece of land. He knows that there is a completely straight pipe underneath some part of his property, but it could be going in any direction. He wants to dig some ditches to find it. He knows that if he digs all around the property we will find it, but that requires 4 miles of digging! What choice of lines minimizes the amount of required digging, and what is that minimum amount?
Just three lines around the perimeter will suffice, cutting the digging down to only 3 miles. After more thought one sees that only 2 diagonal lines are needed, which is 2.82 miles of digging.
I don't think coming up with examples is that hard, it's just that proving it seems impossible. At the moment I found something with length $\sqrt{2}+\sqrt{3/2}$ but I just can't prove it is optimal. It is worth noting that the lines can be disconnected, and can have finitely many pieces. The solution I had above was made up of two different pieces.
Any help is greatly appreciated!