Math Challenge

If you ever had a class with Associate Professor of Mathematics Shahriar Shahriari (now Associate Dean of the College), you may have encountered one or more of these challenge problems in math and logic (minus the corny titles, of course). But do you remember how to solve them? Print this page to work the puzzles, then click here for the solutions.

1. The Case of the Castles and the Careful Queen

A queen wants to build 10 castles connected by ditches. She wants exactly five ditches in the form of straight lines and she wants four castles on every ditch. Her advisors suggest a star-shaped configuration (right) with castles at the points of intersection of lines. She rejects the proposal by adding a new condition. She wants one (or maybe even two) of the castles to be surrounded by ditches and thus not susceptible to direct assault from outside. Can you devise a design that meets all of her requirements?

2. How Many Hands Did Hubby Hold?

A mathematician and her husband go to a party at which there are four other married couples. Some of the 10 people at the party shake hands, but no one shakes hands with his or her own spouse, and so the maximum possible number of hands that any individual can shake is eight. The mathematician asks all nine people at the party how many hands each of them has shaken and receives the nine answers: 0, 1, 2, 3, 4, 5, 6, 7, 8.

How many hands did the mathematician's husband shake?

3. Black Hat, White Hat

Three blindfolded people, Abe, Beth, and Chuck, each take one hat from a barrel containing three black and two white hats. Abe and Beth remove their blindfolds and can each see the hats on the heads of the other two, but not the hat on their own head. Abe says, "I cannot tell what color my hat is." Upon hearing this, Beth says, "I cannot tell the color of my hat either." Hearing these two statements, Chuck, who is still blindfolded, says, "I know what color my hat is." How did he know, and what was the color of his hat?