=Raymond Boute & James Propp [...] the unique feature, which I never saw in any other problem, was that this solution broke down if the basic parameter (possibly some number of persons in the problem statement) reached a certain value that was significantly larger than one would expect for a smallest counterexample (several dozens or perhaps even several hundreds). Does anyone have any idea what problem Raymond is thinking of?
Not specifically, but such breakdowns aren't all that rare--though still fun! I'm reminded, eg, that the coefficients in the factors of x^n-1 are always -1, 0 or +1 until n>=105. There are a number of sequences in the OEIS which first diverge at large n. Search for "differs from", perhaps? Anyone have any favorites?