Here is an interesting puzzle taken from mathproblems.info:
There is a straight cable buried under a unit square field. You must dig one or >more ditches to locate the buried cable. Where should you dig to guarantee >finding the cable and to minimize digging? For example you could dig an X shape >for total ditch length of 2*sqr(2) but there is a better answer.
Let the corners of the square be at (0,0),(1,0),(1,1), and (0,1). The solution is to dig the following ditches:
(0,0) to (x,x); (1,0) to (x,x); (0,1) to (x,x); (1/2,1/2) to (1,1)
Where x = (3-sqr(3))/6 =~ 0.2113248654
The total length of all ditches is aproximately 2.6389584338
How to prove this result? Assuming that the solution is to dig (0,0) to (x,x); (1,0) to (x,x); (0,1) to (x,x); (1/2,1/2) to (1,1), it is straight forward to show that the best value for x is (3-sqr(3))/6, but how to justify that such a digging is optimal?