Re the threads on proofs, and sudoku uniqueness, I thought of a question relating the two threads. Before I ask my question, I'll quote the example from Huang's post : ----------------------- 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 a mathematical solution: Let the radius of A be a and the radius of B be b. Now, call one endpoint of the line segment X, and the tangent point of the line segment to B we will call Y. Call the centers of the two circles Z. Now, since X is on circle A, we know XZ = a. Since Y is on circle B, we know YZ = b. Since YZ is a radius of B and Y is a tangent point of segment XY, we know that angle XYZ is a right angle. Finally, since we know that the extended line going through YZ is a diameter of A and that if a diameter of a circle intersects a chord at right angles then it bisect the chord, XY must be half the length of the chord, to wit, 5. By the Pythagorean theorem on right triangle XYZ, we know that 25 + b^2 = a^2, or 25 = a^2 - b^2. Now, the area of the annulus is the area of A minus the area of B. To wit, the area is pi a^2 - pi b^2, which is equal to pi (a^2 - b^2) = 25 pi. Therefore the area is 25 pi. 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. ---------------------------------- The first proof is a valid proof. If we ran it through a proof-checker, we would get a positive answer "yes this is valid". My question is, is this second proof a valid proof? If we had a proof checking program, and we fed it this proof and asked if it is correct, would it say "yes" or "no" ? If the answer is no, then why do we accept such solutions in exams/puzzle competitions? 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