In order for the second solution to be a valid proof, we need to prove a lemma that the size of the circles doesn't matter. If there's a simple, "proof without words" way of seeing this, then this would be an outstanding solution for a competition or homework. As an aside, I've been musing for many years about the possibility of "automatically" generating "proof without words" types of proofs of certain math theorems. This isn't going to be easy, as Newton himself spent 20 years converting his calculus-style proofs in the Principia into geometric proofs because he knew that no one would be able to understand or verify his calculus style proofs. (Now, of course, most students would find his geometric style proofs _harder_ to follow than the calculus-style proofs!) The basic idea is to reverse the logic of Tarski's decision procedure for geometry, which converts geometry into analytic geometry. Since the mapping is obviously not 1-1, you need a clever way to map questions about polynomials back into questions about lines, planes, circles, etc. At 05:06 AM 3/7/2006, Gary McGuire wrote:
There are two concentric circles, A and B. There is a line segment which is a chord of A and tangent to B. This line has length 10. What is the area of the annulus between the two circles?
There is also a "meta"-solution:
Since this puzzle must have a reasonable solution, the size of the circles are probably irrelevant. Then consider the case where circle B has radius 0. In that case, B becomes a point (the center), the chord becomes a diameter of A, and the annulus becomes the area of A. The question is then: Given that the diameter of a circle has length 10, what is its area? This is easily found to be 25 pi. ---------------------------------- If a student submitted something like this in a puzzle competition (e.g. Putnam or olympiad) or even on a homework, we would all jump up and down with delight at the ingenuity. Well I would. I would be glad to have a student who thinks outside the box. Should I turn around and tell such a student that the argument is not valid? Gary McGuire