You have twelve coins. You know that one is fake. The only thing that distinguishes the fake coin from the real coins is that its weight is imperceptibly different. You have a perfectly balanced scale. The scale only tells you which side weighs more than the other side.
What is the smallest number of times you must use the scale in order to always find the fake coin?
Use only the twelve coins themselves and no others, no other weights, no cutting coins, no pencil marks on the scale. etc.