CS2984: Introduction to Problem Solving
Homework Assignment 11

Due at 11:00pm on Wednesday, November 4
30 Points

See the General Guidelines for homework assignments. Don't forget to include the pledge statement!

This assignment may optionally be done with a partner. You are strongly advised to use a partner. If you do use a partner, then:

Partners must work together on the problems, not independently.
The final submission must contain the name of both partners.
Only one partner will make the actual submission to Web-CAT.
Your submission must explain, for each problem, the contributions of each partner.

Bottle A contains a quart of milk, and Bottle B contains a quart of coffee. (Note that the bottles themselves are as big as needed.) Pour some amount of B into A, mix well, then pour this amount back into B until both bottles contain a quart of liquid. What is the relationship between fraction of coffee in A and the fraction of milk in B? You must explain your answer.
Problem (that we did in class): Assume that every point in the plane is colored either red or blue. Prove that there exist two points that have the same color and that are exactly 1 mile apart.
Attempted argument:
1. Take an equilateral triangle with sides 1 mile in length.
2. By the Pigeonhole Principle, there must be two points having the same color that are exactly 1 mile apart.
This argument obviously is missing some steps. Expand on this to make a complete argument.
What is the next letter in the sequence O, T, T, F, F, S, S, E, ...?
What is the rule for obtaining the next letter?
Find the next member in this sequence: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
What is the rule for obtaining the next member?
Pat wants to take a 1.5-meter-long sword onto a train, but the conductor won't allow it as carry-on luggage. And the baggage person won't take any item whose greatest dimension exceeds 1 meter. What should Pat do?