Oh, this is an easy one. You develop a casino game that is based on the players somehow competing against the inevitability of hitting 1 eventually. This will work for practically every possible case, but the cheaters are clever. Given the "I want to beat the house" incentive, they will come up with a counter-example (should one exist) just to beat you. In the case it does exist, you will have provided the circumstances to disprove the conjecture, and otherwise you will make a lot of money. Simple!
