Two Math Riddles

Usually I am not particularly good at telling math riddles or solving them, and I was not planning on presenting math riddles here. But these days, when mathematical blogs break new ground and enter uncharted territories, let me make an exception and tell you two riddles.

1) (Told to me by Sergiu Hart; taken from a MSRI news-teller to the best of my memory.)

You have a 1 meter long (one ant thin) wooden stick and you put on it n ants in arbitrary locations. Each ant is facing one end of the stick. The ants move 1 meter per minute and when two ants meet they both start moving in the opposite directions. If an ant reaches an end of the stick it falls from it.

After how much time you can be sure that no ants will remain on the stick?

2) (Told to me (with the solution) most recently by Sasha Barvinok, who heard the problem on “Car talk“.)

An airplane has 100 seats, and 100 passengers were assigned seats. The first passenger, ‘Joe,’ enters the plane and rather than sitting in his assigned place, he sits in a random place. The next passengers come one by one and every passenger sits in his assigned seat if it is empty, and in a random empty seat if his seat is already taken. What is the probability that the last passenger ‘Jim’ will sit in his assigned seat?

If you want to solve these riddles on your own do not read the rest of this entry. Continue reading