Problem with rectangle

 If a finite number of rectangles, every one of which has at least one integer side, perfectly tile a big rectangle, then the big rectangle also has at least one integer side.

An interesting proof

roof by complex integration

A standard proof of theorem 1 (IBM 1999b) involves integrating the complex function e^2πi (x+y) over the two-dimensional plane. The integral over any rectangle is zero if and only if at least one side is an integer; so the integral over each small rectangle is zero; the integral over the big rectangle is the sum of the small rectangles' integrals, so it equals zero too; therefore the big rectangle has an integer side.

This proof is probably not accessible to most ten-year olds; nor is the second solution given at (IBM 1999b), which involves large prime numbers. And neither of these proofs helps prove theorem 3.

I now present two proofs of theorem 1, both accessible to ten-year-olds. The first proof is similar in character to the complex integral proof sketched above. The second proof uses a completely different approach, which can be applied immediately to theorem 3. Both proofs have been published before (Wagon, 1987).

Figure 1. Illustration of the checkerboard proof. Dashed blue lines are an integer grid. The checkerboard squares are of size 1/2. In the lefthand example (a), the central small rectangle violates the constraint of the theorem: it has both sides non-integer. In the righthand example (b), every small rectangle has one side of integer length. Every small rectangle covers equal amounts of black and white, so the large rectangle must do the same.