My comment is a joke as well. If you frame Buffon's needle game as a Martingale, playing until you get an answer that agrees with Fibonacci's previous work does not meet the preconditions for the optional stopping theorem. Intuitively, it's equivalent to starting with $1 and making fair bets with infinite credit until you have $2. Sure, it's possible, but it's not a "fair" betting game.

I just thought the quote from the article was unbelievably good -- "choosing his stopping point judiciously" is such a well-done understatement.