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
.
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 , 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 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 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.