Four farm-hands need to carefully walk the entire perimeter of a large square field to check for signs of infestation. They can each walk separately and any section of edge need only be checked by one person. The field is 200m on an edge and they all start at the same corner. Each person can either walk normally at 2m/sec or walk while checking at only 1m/sec. Any person may cut through part of the field at a normal walking pace. They must all finish at the opposite corner of the field.
What is the shortest time in which they can check the entire edge?

Consider the same problem with a circular field of radius 100m.
How long would this take?

(In reply to this is probably the best (fastest) strategy by Charlie)

I agree with Charlie's solutions.  Since they all have to end up at the same spot, they all need to check 1/4 of the total distance in order to minimize the time it takes for all four men to arrive.  If one of them has to check more than 1/4, it will just slow down the whole process of getting them all to the opposite corner.

Starting from the southeast corner, you could find a path that would get some of the farm hands there faster, but someone's fastest path will be to walk north to the northeast corner, then west to the northwest corner (or west and then north, by symmetry).  No matter which way you slice it, this person will walk half this distance at 2m/s and half at 1m/s, so even if someone finishes sooner, the earliest that all four will be done is 300 s.

This idea is exactly the same for the case of the circle - each man must cover 1/4 of the edge.  Charlie's solution clearly minimizes the time this will take.

Posted by tomarken on 2006-04-20 11:29:35