“The Handshake Problem”

Problem SolvingCh 1

Shake Hands with Everyone

• Some things to think about:

How many handshakes occurred?

How did you keep track that you didn’t shake someone’s hand twice?

How do you know you shook everyone’s hand?

Is Your Friend a Cheater?

You’re friend invited you to play a game

You get 2 darts where there are six sections scored 1, 2, 3, 4, 5, and 6.

The rule is you throw both darts. If the second score is greater than the first, you win. Otherwise you loose.

Is the game fair?

Review• There are many different approaches to a

problem. And there is not one that is better than another. This chapter will focus on trying different methods.

• The first part of your homework will focus on solving problems similar to the handshake problem.

• The second part of you homework will focus on solving problems with a graphical model.

Day 2 - Geometric Modeling

• Let’s look at the handshake problem again.

• Can anyone think of how to draw a geometric figure to represent our class size and the number of hand shakes?draw a shape with the # of

sides = to the # of students. vertices = ppltotal # of diag + the # of sides = # of handshakes

Practice - on your own

1.How many diagonals are there in an octogon?

2.An icosagon (20 sides) has how many diagonals?

3.If there were 35 people in this class, how many handshakes would be required if everyone were to shake hands?

One of the most important mathematical

tools:

Alwaysdraw

a picture!

Finding a Pattern

We’re in Algebra now right?!

So let’s look at the handshake problem algebraically...

Who knows what sigma notation is?

Sigma Notation

Let’s find a pattern for the total number of handshakes for different

class sizes.

2 people = 1 handshake

3 people = 3 handshakes

4 people = 6 handshakes

5 people = 10 handshakes ...

•What is the pattern in the total number of handshakes?

Pattern:

• Below is the sequence representing the total number of handshakes:

• 1, 3, 6, 10, ...

Now here is the series of how the handshakes are increasing when another person is added to the class...

1 + 2 + 3 + 4 + ...

Sigma Notation• This needs to represent a sum = the total number of

handshakes.

• Let n = the number of people in the class.

• i.e. when n = 2, there was 1 handshake.

when n = 3, there was 2 handshakes added to the previous amount.

• Is there an expression representing the number of people and the number of handshakes added to the total?

n - 1

What would the sigma notation look like for 8 people in the class? Then evaluate

it.

• Remember:

n 1n2

8

(2-1) + (3-1) + (4-1) + (5-1) +

(6-1) + (7-1) + (8-1) = 28

Day 3 - A Recursive Solution

• Recursive - a rule in which determining a certain term in a sequence relies on the previous term(s).

Handshake Problem (again)

Yesterday, we looked at how the total number of handshakes were increasing with each person added to the room:

•1, 2, 3, 4, ...

•Let’s think of it a different way:

Handshake Problem (again)

With 2 people, there is 1 handshake

With 3 people, 2 more handshakes were added

With 4 people, 3 more handshakes were added

• Therefore: H2 1H 3 H2 23H 4 H 3 36

Handshake Problem (again)

• This is a recursive sequence.

• Can you think of a way to write it algebraically?

Hn Hn 1 n 1

Something important is

missing!

Checkpoint

1.Generate the first five terms of the sequence that has the recursive definition

2.Find a recursive formula that will generate the sequence 1, 2, 6, 24, 120, 720

T1 2,Tn 3Tn 1

Demonstrations

• 1st - Think of an expression that represents an even number

• 2nd - Think of an expression that represents an odd number

Even and Odd Integers

• The even and odd integers can be divided into two sets with no intersection, called disjoint sets.

• a) Draw a picture to represent this situation

• Keep your answer on a whiteboard and wait to share with other groups.

Even and Odd Integers

• Prove it!

• I will give each group one of the problems (b - g) to prove algebraically.

• Hint: even integers are represented as 2n and odd integers are represented as 2n - 1

Even and Odd Integers

• Which set of integers contain the answer to each of the following operations:

• b) Add two even integers

• c) Add two odd integers

• d) Add an even and an odd integer

• e) Multiply two even integers

• f) Multiply two odd integers

• g) Multiply an odd and an even integer

• Keep your answer on one of your whiteboards