The Fake Coin
This puzzle is an old problem. You are given n coins, of which n-1 are genuine and one is fake. All genuine coins have the same weight and the fake coin has a different weight. The difference is only perceptible by using a balance. Find the fake coin and tell if it is lighter or heavier in m weighings as maximum.
The game has three levels:
n=3 coins, m=2 weighings
n=12 coins, m=3 weighings
n=39 coins, m=4 weighings