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 A faster strategy (spoiler) by Steve Herman)

Excellent idea.

If x is the distance from the corner that the first farmhand leaves the fence and cuts across the field, then

t1 = 2*(200-x)/2 + 2x = 200+x

t2 = 2*(200-x) + sqrt(2*x^2)/2 = 400-2x+sqrt(x^2/2)

Equating the two times to find the minimum yields

x = 87.23
t_min = 287.23

Posted by Bob Smith on 2006-04-20 13:35:43