Math 160 Mathematical Problems and Games, Fall 2002
Problem Set 2: hand in one problem Tuesday, Sept. 24.

Instructions:  Think about all four problems, and come up with some ideas about how to solve them. Then choose one of the problems and write up a careful solution. Your solution should be written so that it would convince a skeptical classmate.

1. You have seven glasses on a table, all face down. You are allowed to turn over four glasses at a time, as many times as you want. Is it possible to end up with all seven glasses face up?

2.  You have 9 coins, one of which is counterfeit and weighs less than the others.  Use two weighings on a balance with two pans to find the counterfeit coin.

3. You have 8 coins, one of which is counterfeit. You do not know whether the counterfeit coin is heavier or lighter than the others.  Use three weighings on a balance with two pans to find the counterfeit coin.

4.  You have 12 coins, one of which is counterfeit. You do not know whether the counterfeit coin is heavier or lighter than the others.  Use three weighings on a balance with two pans to find the counterfeit coin.

5.  The numbers 1 through 10 are written in a row.  Can the signs "+" and "-" be placed between them in such a way that the resulting expression adds up to 0?

6.  Twenty-five men and twenty-five women are seated at a round table.  Show that there must be at least one person at the table who is seated between two men.