Ok, so this guy I know posed me this question. Suppose you have a rectangle that’s divided into a finite number of smaller rectangles which have the property that at least one of the sides has integer length. Prove that the large rectangle has the same property. He says he know of 3 separate proofs, one using calculus, one using the fact that there are infinite primes, and one just using basic geometry. I was looking to attack it using induction. Any idea on any of the proofs?
Would that be:
at least one of the sides of ONE of the sub-rectangles has integer length OR
at least one of the sides of ALL of the sub-rectangles have integer length OR
at least one of the sides of AT LEAST ONE of the sub-rectangles have integer length?
The second one.
I just tossed this one to a couple of our first-year grad students a couple months ago. The calculus solution is the best, and once you see it you’ll understand why you never need to see another.
As for a hint, how could you tell if a rectangle has at least one integer-length side using calculus? Something that would apply to a rectangle as well as to the union of a bunch of rectangles.