In front of you, there are 9 coins. They all look absolutely identical, but one of the coins is fake. However, you know that the fake coin is lighter than the rest, and in front of you is a balance scale. What is the least number of weightings you can use to find the counterfeit coin?
10 people came into a hotel with 9 rooms and each guest wanted his own room. The bellboy solved this problem.
He asked the tenth guest to wait for a little with the first guest in room number 1. So in the first room, there were two people. The bellboy took the third guest to room number 2, the fourth to number 3, ..., and the ninth guest to room number 8. Then he returned to room number 1 and took the tenth guest to room number 9, still vacant.
How can everybody have his own room?
Suppose we lay down two cups in front of you. One of the cups is filled with tea and the other one with coffee. Now we ask you to take a spoonful of tea and mix it with the coffee. At this moment, the coffee cup has a mixture of tea and coffee. You have to take that mixture (spoonful) and add it back to the tea.
Can you now tell if the cup of coffee has more tea or the cup of tea has more coffee?