Gary 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 a mathematical solution: . . . There is also a "meta"-solution: Since this puzzle must have a reasonable solution, the size of the circles are probably irrelevant. . . . . . . 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?

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Are you sure that such solutions *are* given full credit in Putnam exams and Olympiads? I don't know if that's the case. I would definitely *not* give full credit for such a solution, though I wouldn't deduct a lot, either. The ideal student would use the absence of information as a tip-off that the answer most likely is independent of the missing info, and then prove this fact as part of the solution. (It's really not that hard to label a diagram and then write r^2 + 5^2 = R^2, so pi*R^2 - pi*r^2 = pi*(R^2 - r^2) = pi*25.) --Dan