Math 5336 Number Theory Dr. Cordero

Class activities
for August 26^{th}, 2002

Topic: Mathematical Induction

1.
**The Tower of Hanoi
Problem**:

(Hand-out #1)

*
There are three posts. On one post
(A) there is a stack of disks with different diameters. The problem is to
transfer all the disks to one of the other posts (B or C) in the fewest possible
moves, moving one disk at a time in such a way that no disk is ever placed on
top of a smaller disk.*

__
__

The Tower of Hanoi problem is:

*
What is the smallest number of
moves required to transfer 64 disks from one post to another following the
instructions?*

*
*

a. Solve the Tower of Hanoi problem.

b. Work Problems 1-5 on Hand-out # 1.

**
2.
The Envelopes
Problem:**

*
(Hand-out # 2)*

*
A certain store sells envelopes in
packages of five and packages of twelve and you want to buy n envelopes. Prove
that for every , this store can fill an order for exactly n envelopes
(assuming an unlimited supply of each type of enevelope package).*

*
*

a. Solve the problem.

b. Work on Exercise # 3 on p.159.

**
3.
Mathematical
Induction**

(Hand-out # 3)

a. Study the principle of Mathematical Induction.

b. Study the proof on page 9 of the following;

For every positive integer ,

c. Complete the proof on page 10.

**
4.
Additional Practice**

a. Hand-out #2: Exercises # 5, 6 a, 6c, 12

b. Textbook, Section 1.1, pp.6-7 # 1

Topic for next
class: * Divisibility Theory in the Integers*, Chapter 2 in the
textbook.