i) The fake coin
A rather intriguing riddle which pops up in different mathematics-related interviews in a more simple form is the following:
You have twelve coins. You know that one is false. The only thing that distinguishes the fake coin from the real one is that its weight is imperceptibly different, but not necessarily lighter or heavier. You have a perfectly balanced scale, but it 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?
ii) The 100 coins
This one is easier, but has a creative solution:
There are 10 sets of 10 coins. You know how much the coins should weigh. You know that all the coins in one set of ten are exactly a tenth of a gramme off, making the entire set of ten coins a gramme off. You also know that all the other coins weight the correct amount. You are allowed to use an extremely accurate digital weighing machine only once. How do you determine which set of 10 coins is faulty?