Problem of the Month for April, 2006

Problem:

The house where Gina lives is equipped with an elevator. The house has four floors, including the basement and the attic. The elevator is very unusual: it has only two buttons, .A. (toward the attic) and .B. (toward the basement) and goes only one floor up or down at a time. It takes 30 seconds between the time you press the button until the elevator reaches the corresponding floor above or below (at which time you have to press a button again). Furthermore, the elevator doesn.t work if you press two consecutive identical strings of two or more buttons. For example, suppose you start in the basement and proceed with the sequence AAABBAB. Then the elevator willl stall when you next press a button: for if you press A, you have BA two times in a row, and if you press B, you have ABB two times in a row. If she starts in the basement, can Gina ride the elevator for more than 8 minutes before she is stymied by a stalled elevator? What if she starts on the first (ground) floor? Explain.

Possible Generalizations to consider:

If Gina starts in the basement and rides until the elevator stalls, find all the different rides she could make. If you think you have found them all, explain why you think you are right. How long can Gina go before stalling? What if there are five floors? Six floors?