Farmers Field
Problem
A farmer has a flat field and two sons who will each inherit half of the field. The farmer wishes to build a stone wall to divide the field in two so each son inherits the same area. Stone walls are expensive to build, so naturally the farmer wishes to build the shortest wall he can.
Can you prove that the shortest wall is always straight, whatever the shape of the original field? Or perhaps you can find a shape where the shortest wall isn't straight!
Student Solutions
Phil of Westwood High School, Leek, Staffordshire wrote:
The problem called farmers fields in the 15+ challenges can be proven wrong, the shortest wall isn't always a straight line.Consider this field: -
If the left side is slightly larger than the right side then the shortest distance to build the wall could be to use a curve as shown by the red line (A).To use straight lines would involve using three if placed as shown by the green lines (B), or two straight lines could be used as shown by the purple line (C).
To use one straight line the shortest route would be as indicated by the blue line (D).It is obvious from looking that when the difference is very small that a single straight line won't be the shortest difference.