Four farmhands need to walk the entire perimeter of a very large square field to check the fence posts. They can each walk separately but every section of edge needs to be walked by at least one person. The field is 200m on an edge and they all start at the same corner and can walk at 2m/sec. They do not need to finish at the same place and they may cut through part of the field if necessary (and they walk at the same pace if they do).
What is the shortest time in which they can check the entire edge?
How long would it take if there were only 3 farmhands?
Consider the same problem with a circular field of radius 100m.
How long would it take to check the entire perimeter with 4 farmhands?
With only 3?
Since we don't want any wasted time, it wouldn't seem to make sense to have any of the farmers walk over the same part of an edge. If they all start in the southeast (bottom right) corner, we can send one (Farmer A) north up the right edge, and have another (Farmer B) walk northwest across the field to the top edge. From there he will walk west to the northwest corner. We want Farmer B to start at a point on the edge where Farmer A will finish up. Since we don't want to waste any time, the distance Farmers A and B walk should be equal.
The path Farmer A walks will be 200 + x meters. The path Farmer B walks will be (200  x) + sqrt(200^2 + x^2). Then:
200 + x = 200  x + sqrt(200^2 + x^2)
2x = sqrt(200^2 + x^2)
4x^2 = 200^2 + x^2
3x^2 = 40000
x^2 = 13333.33
x = 115.47m
So Farmer A will walk 200m up the right edge, then 115.47m across the top edge. Farmer B will walk 230.94m across the field to this point and walk the remaining 200  115.47 = 84.53m to the northwest corner. They will both walk 315.47m which at 2m/s will take 157.735 s or about 2 minutes and 37.7 seconds.
Note that at the same time, by symmetry, Farmers C and D will be doing the same exact thing on the other two sides of the field. I don't know if this is the shortest time possible, but it seems like it would be since there is no time "wasted" by walking over the same path or waiting for another farmer to "catch up".

Posted by tomarken
on 20060418 10:10:47 