CS 381 Homework 1


Hand in 1, 2 and 3 of the following questions:

1. Given a 9 pints pail and a 16 pints pail, find a way to get 1 pint in the 16 pint pail.
Try "Working Backward" heuristic.

2. In a party with 100 people, among any set of four there is at least one person who knows each of the other three. There are three people who are not mutually acquainted with each other(that is none of the three knows any of the other two). Prove that the other 97 people know everyone at the party. (Assume that if A knows B, then B also knows A.)

Try "Contradiction" heuristic.

3. Prove that $ \mid \mid x \mid - \mid y \mid \mid \leq \mid x - y \mid $ .
Try "Divide into Cases" heuristic.


The questions below are for your interest. You do not have to hand them in. Solutions to them are going to be posted after you hand in homework 1.

4. Prove that tex2html_wrap_inline28 , where x and y are positive real numbers.
Try "Working Backward" heuristic.

5. Prove that the sum of two odd squares (i.e. the square of a number which is odd) cannot be a square.
Try "Contradiction" heuristic.

6. Let tex2html_wrap_inline34 be a set of integers such that if any of them is removed, the remaining ones can be divided into two sets of n integers with equal sum. Then prove that tex2html_wrap_inline38 s are either all even or all odd.
Try "Contradiction" heuristic.


Due July 3, 2006, end of the day

You may discuss these questions among yourselves and/or with me. But you must write the answers in your own words.