Problem:
A fenced, rectangular field measures meters by meters. An agricultural researcher has meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the meters of fence?
Solution:
Suppose that the field is partitioned into squares of side . Then there are positive integers with
Hence
so there is a positive integer with and . Note that the total number of test plots is a maximum when is as large as possible. The total length of fence used in partitioning the field into squares is
Since at most meters of fence can be used, we have
so that
Since must be an integer, the largest possible value of is . For this value of we have a total of
squares, formed by using meters of the available meters of fence.
The problems on this page are the property of the MAA's American Mathematics Competitions