Michael Kleber wrote:
The answer to Gary's "is this a valid proof" question, and the sudoku question to which he's drawing the analogy, both boil down to asking what your set of axioms is. In calculating the area of the annulus, Gary's second proof is valid only if you assume that the answer is independent of the radii -- which is, perhaps, an application of the pseudo-axiom "there is something clever which makes this problem worth asking", a frequent domain in which to solve olympiad-style problems.
This clears it up. I assume that a proof checking program will only use ZF, so would declare the second proof invalid. Dan wrote:
Are you sure that such solutions *are* given full credit in Putnam exams and Olympiads? I don't know if that's the case.
No, I'm not sure. I thought there would be some readers of this group who know. I know this particular problem is easy to solve the other way, but I've seen tricky problems which are made much easier by reasoning "here I am sitting in a Putnam/whatever, therefore this problem has a nice solution, therefore....." Dan wrote:
Suppose a sudoku has *exactly* 2 solutions. Then what is the maximum number of boxes that can differ in the two solutions?
502008471040710203107000000300104607070000105010057300601002730720030014493871500 532968471849715263167243859385124697276389145914657382651492738728536914493871526 562398471948716253137425986359184627874263195216957348681542739725639814493871562 42 clues, 39 positions different in the two solutions. I'm sure one can do better. Gary McGuire