1 [easy]. Suppose you have 100 bags of coins, each containing at least 363 coins. A genuine coin weighs 10 grams. One bag is full of counterfeit coins, which are a gram lighter. If you have a weighing machine that shows the weight of its load, how can you work out in one weighing which bag has counterfeit coins in it?
2 [harder but fun; this one shows how information propagates]. You have a balance and 10 coins. One of them is counterfeit but you do not know if it is lighter or heavier than the genuine coins. Work out in three weighings which coin is counterfeit.
There is a legend that at a crucial point in WW2 the code-breakers of Bletchley Park became so obsessed with this kind of puzzle that somebody proposed dropping copies of them behind enemy lines in the hope that German intelligence officers would become equally obsessed and their efficiency would also be impaired. Sadly, I’ve been unable to track down a reliable source for this story… (DP)