I assume that you know that an Italian mathematician by the name of Lazzarini published in the early twentieth century an estimate of pi that he claimed to have obtained from a Buffon’s needle experiment with 3408 needle drops and getting the estimate pi ~ 355/113 (which is a well-known rational approximation). If he had done one more drop, then irrespective of what the outcome was, his estimate would have been much worse. This has been cited as a well-known case of academic cheating.