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 the square field, the first farmer inspects a fence for a distance of approximately 112.774 meters, turns 45 degrees to cross the inner field at a normal walk, then completes the inspection to the opposite corner for a distance of approximately 112.774 meters, the trek would take approximately 287.266 seconds. The second farmer can walk normal 112.774 meters along the fence then inspect the next approximate 174.452 meters (turning at the corner) with a final approach inspection for the approximately final 112.277 meters for a total of 287.266 seconds.  The third and fourth farmers can provide symmetry with first two farmers by inspecting the other two fences in the same period of time.

In the circular field, the first farmer inspects the fence for one-quarter of the circumference of the field then bee-lines it to the final meeting point. The second farmer crosses the field straight to the point where the first farmer finished his inspection and inspects the next quarter.  The third and fourth farmers, again, provide symmetry to the first two. The time for all four farmers to complete their inspection takes approximately 227.790 seconds [50*(SQRT(2)+PI)].

Posted by Dej Mar on 2006-08-01 05:25:42