Class Problems, Lecture 1

Class Problems, Lecture 1
On occasion I’ll start class by writing one or two problems on the board. You’re expected to
come in, if you like find one or two people to work with, and see if you can answer the problems.
Then we’ll talk about them a bit. You may be asked to present your solution.
If you’ve seen a problem before, don’t wreck it for others by telling them the solution. Let me
know if you want another problem.
Here are problems for today. Find someone to work with and start on them.
Problem 1
After lunch one day, Alice suggests to Bob the following method to determine who pays. Alice pulls
three six-sided dice from her pocket. These dice are not the standard dice, but have the following
numbers on their faces:
• Die A: 1,1,6,6,8,8
• Die B: 2,2,4,4,9,9
• Die C: 3,3,5,5,7,7
The dice are fair, so each side comes up with equal probability. Alice explains that each of them
will pick up one of the dice. Then they will each roll their die, and the one who rolls the lowest
number loses and will buy lunch. So as to take no advantage, Alice offers Bob the first choice of
the dice. What should Bob choose?
Problem 2
On a different day after lunch, Bob suggests a different game to see who pays. Alice on Bob will
each choose a different sequence of three flips. (So they could choose “Heads-Tails-Heads”, or
“Tails-Tails-Tails” for example.) After they choose, a fair coin will be tossed until one of their
sequences appears as a consecutive subsequence of the coin tosses. The player whose sequence
appears first wins. Bob says Alice can choose first, and after she chooses and tells him her sequence
he’ll choose a different sequence. What should Alice choose?