Problem:
A block of cheese in the shape of a rectangular solid measures by by . Ten slices are cut from the cheese. Each slice has a width of and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic of the remaining block of cheese after ten slices have been cut off?
Solution:
Let , and be the dimensions of the cheese after ten slices have been cut off, giving a volume of . Because each slice shortens one of the dimensions of the cheese by . By the Arithmetic-Geometric Mean Inequality, the product of a set of positive numbers with a given sum is greatest when the numbers are equal, so the remaining cheese has maximum volume when . The volume is then .
Query: What happens if is not divisible by ?
The volume of the remaining cheese is greatest when it forms a cube. This can be accomplished by taking one slice from the dimension, slices from the dimension, and slices from the dimension for a total of slices. The remaining cheese is then by by for a volume of .
The problems on this page are the property of the MAA's American Mathematics Competitions