Math 5336 Number Theory                                                       Dr. Cordero

 

Class activities  for August 26th, 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.