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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.

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?

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  • $\begingroup$ Do we have any information about where the cable is? It could be coiled up in a tight ball, so tight it can be modelled as a single point, and placed in a random location. If so, we have no choice but to dig up the whole field. Does it stretch across the field? If so, in which direction? Also, what is meant by "ditch"? From the solution, I'm guessing a ditch is a line segment? $\endgroup$ – Jack M Mar 21 '14 at 12:40
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This is "The Opaque Square Problem". http://arxiv.org/abs/1311.3323 is a recent paper about it, with references.

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