Patterns and Growth

April 2002 TM MATH: Patterns & Growth 1

Patterns and Growth

John Hutchinson


Problem 1: How many handshakes?

Several people are in a room. Each person in the room shakes hands with every other person in the room. How many handshakes take place?


People Handshakes

1 0

2 1

3

4

5

6

7


People Handshakes

1 0

2 1

3 3

4

5

6

7


People Handshakes

1 0

2 1

3 3

4 6

5

6

7


People Handshakes

1 0

2 1

3 3

4 6

5 10

6 15

7 21


Is there a pattern?


Here’s one.

People Handshakes

1 0 0

2 1 1

3 3 1 + 2

4 6 1 + 2 + 3

5 10 1 + 2 + 3 + 4

6 15 1 + 2 + 3 + 4 + 5

7 21 1 + 2 + 3 + 4 + 5 + 6


Here’s another.

People Handshakes

1 0 0

2 1 1 + 0

3 3 2 + 1

4 6 3 + 3

5 10 4 + 6

6 15 5 + 10

7 21 6 + 15


What is:

1 + 2 + 3 + 4 + …..+ 98 + 99 + 100?


Look at:

1 2 3 4 … 98 99 100

100 99 98 97 … 3 2 1

101 101 101 101 … 101 101 101

There are 100 different 101s. Each number is counted twice. The sum is

(100*101)/2 = 5050.


Look at:

1 + 2 + 3 + 4 + 5 + 6 = 3 7 = 21

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 4 7 = 28


If there are n people in a room the number of handshakes is

n(n-1)/2.


Problem 2: How many intersections?

Given several straight lines. In how many ways can they

intersect?


2 Lines

1 0


3 Lines

0 intersections 1 intersection

2 intersections 3 intersections


Problem 2A

Given several different straight lines. What is the maximum number of

intersections?


Is the pattern familiar?

Lines Intersections

1 0

2 1

3 3

4 6

5 10


Problem 2B

Up to the maximum, are all intersections possible?


What about four lines?


What about two intersections?


What about two intersections?

Need three dimensions.


Problem 3

What is the pattern?

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…


Note

• 1 + 1 = 2

• 1 + 2 = 3

• 2 + 3 = 5

• 3 + 5 = 8

• 5 + 8 = 13

• 8 + 13 = 21

• 13 + 21 = 43


This is the Fibonacci Sequence.

Fn+2 = Fn+1 + Fn


Divisibility

1. Every 3rd Fibonacci number is divisible by 2.

2. Every 4th Fibonacci number is divisible by 3.






Sums of squares

12 + 12 1 2

12 + 12 + 22 2 3

12 + 12 + 22 + 32 3 5

12 + 12 + 22 + 32 + 52 5 8

12 + 12 + 22 + 32 + 52 + 82 8 13


Pascal’s Triangle

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1


1 = 1

1 1 = 2

1 2 1 = 4

1 3 3 1 = 8

1 4 6 4 1 = 16

1 5 10 10 5 1 =32


Note

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 1 2 3 5 8


Problem 3A: How many rabbits?

Suppose that each pair of rabbits produces a new pair of rabbits each month. Suppose each new pair of rabbits begins to reproduce two months after its birth. If you start with one adult pair of rabbits at month one how many pairs do you have in month 2, month 3, month 4?


Let’s count them.

Month Adults Babies Total

1 1 0 1

2 1 1 2

3 2 1 3

4 3 2 5

5 5 3 8

6 8 5 13


Problem 3B: How many ways?

A token machine dispenses 25-cent tokens. The machine only accepts quarters and half-dollars. How many ways can a person purchase 1 token, 2 tokens, 3 tokens?


Lets count them.

Q = quarter, H = half-dollar

1 token Q 1

2 tokens QQ-H 2

3 tokens QQQ-HQ-QH 3

4 tokens QQQQ-QQH-QHQ-HQQ-HH 5

5 tokens QQQQQ-QQQH-QQHQ-QHQQ

HQQQ-HHQ-HQH-QHH

8


2 3

5

813

C D E F G A B C

Observe


Observe

• C 264

• A 440

• E 330

• C 528

• 264/440 = 3/5

• 330/528 = 5/8


Note

144

89

89

55






Flowers

# Petals Flower Flower Flower1 White Calla Lily

2 Euphorbia

3 Euphorbia Lily Iris

5 Columbine Buttercup Larkspur

8 Bloodroot Delphinium Coreopsi

13 Black-eyed Susan

Daisy Marigold

21 Daisy Black-eyed Susan

Aster

34 Daisy Sunflower Plantain


References

Documents

Patterns and Growth