Ontario Mathematics Olympiad

Sample Question #5

Alice in Wonderland

Lewis Carroll, the author of the story of Alice in Wonderland was not only a very talented writer, but he was quite fond of mathematical puzzles. The following puzzle has been attributed to him:

A queen and her son and daughter are being held in the top room of a high tower. Outside their window is a pulley with a rope over it, and a basket at each end of the rope. The baskets are of equal weight. The one outside the window is empty, and the other on the ground contains a stone with a mass of 30 kilograms. The stone serves as a counterweight.

There is enough friction in the pulley so that it is safe for anyone to be lowered in one basket provided his/her mass is not greater than the mass of the other basket by more than six kilograms. If the difference is greater than six kilograms, they might come down with such a speed that the bump at the bottom might injure them. Of course, when one basket goes down, the other basket goes up to the window.

The queen's mass is 78 kilograms, the daughter's mass is 42 kilograms, and the son's is 36 kilograms. What is the simplest algorithm (fewest number of steps) by which they all can get to safely to the ground? The basket is large enough to hold any two people, or one person and the stone. No one assists the prisoners in escaping nor can they help themselves by pulling on the rope. In other words, the pulley operates only when the mass in one basket exceeds the mass in the other.

Assuming the last person gets out of the way quickly, make up a diagram of the movements (___________ down; __________ up).