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April 2002 TM MATH: Patterns & Growth 1
Patterns and Growth
John Hutchinson
April 2002 TM MATH: Patterns & Growth 2
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?
April 2002 TM MATH: Patterns & Growth 3
People Handshakes
1 0
2 1
3
4
5
6
7
April 2002 TM MATH: Patterns & Growth 4
People Handshakes
1 0
2 1
3 3
4
5
6
7
April 2002 TM MATH: Patterns & Growth 5
People Handshakes
1 0
2 1
3 3
4 6
5
6
7
April 2002 TM MATH: Patterns & Growth 6
People Handshakes
1 0
2 1
3 3
4 6
5 10
6 15
7 21
April 2002 TM MATH: Patterns & Growth 7
Is there a pattern?
April 2002 TM MATH: Patterns & Growth 8
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
April 2002 TM MATH: Patterns & Growth 9
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
April 2002 TM MATH: Patterns & Growth 10
What is:
1 + 2 + 3 + 4 + …..+ 98 + 99 + 100?
April 2002 TM MATH: Patterns & Growth 11
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.
April 2002 TM MATH: Patterns & Growth 12
Look at:
1 + 2 + 3 + 4 + 5 + 6 = 3 7 = 21
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 4 7 = 28
April 2002 TM MATH: Patterns & Growth 13
If there are n people in a room the number of handshakes is
n(n-1)/2.
April 2002 TM MATH: Patterns & Growth 14
Problem 2: How many intersections?
Given several straight lines. In how many ways can they
intersect?
April 2002 TM MATH: Patterns & Growth 15
2 Lines
1 0
April 2002 TM MATH: Patterns & Growth 16
3 Lines
0 intersections 1 intersection
2 intersections 3 intersections
April 2002 TM MATH: Patterns & Growth 17
Problem 2A
Given several different straight lines. What is the maximum number of
intersections?
April 2002 TM MATH: Patterns & Growth 18
Is the pattern familiar?
Lines Intersections
1 0
2 1
3 3
4 6
5 10
April 2002 TM MATH: Patterns & Growth 19
Problem 2B
Up to the maximum, are all intersections possible?
April 2002 TM MATH: Patterns & Growth 20
What about four lines?
April 2002 TM MATH: Patterns & Growth 21
What about two intersections?
April 2002 TM MATH: Patterns & Growth 22
What about two intersections?
Need three dimensions.
April 2002 TM MATH: Patterns & Growth 23
Problem 3
What is the pattern?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…
April 2002 TM MATH: Patterns & Growth 24
Note
• 1 + 1 = 2
• 1 + 2 = 3
• 2 + 3 = 5
• 3 + 5 = 8
• 5 + 8 = 13
• 8 + 13 = 21
• 13 + 21 = 43
April 2002 TM MATH: Patterns & Growth 25
This is the Fibonacci Sequence.
Fn+2 = Fn+1 + Fn
April 2002 TM MATH: Patterns & Growth 26
Divisibility
1. Every 3rd Fibonacci number is divisible by 2.
2. Every 4th Fibonacci number is divisible by 3.
3. Every 5th Fibonacci number is divisible by 5.
4. Every 6th Fibonacci number is divisible by 8.
5. Every 7th Fibonacci number is divisible by 13.
6. Every 8th Fibonacci number is divisible by 21.
April 2002 TM MATH: Patterns & Growth 27
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
April 2002 TM MATH: Patterns & Growth 28
Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
April 2002 TM MATH: Patterns & Growth 29
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
April 2002 TM MATH: Patterns & Growth 30
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
April 2002 TM MATH: Patterns & Growth 31
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?
April 2002 TM MATH: Patterns & Growth 32
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
April 2002 TM MATH: Patterns & Growth 33
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?
April 2002 TM MATH: Patterns & Growth 34
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
April 2002 TM MATH: Patterns & Growth 35
2 3
5
813
C D E F G A B C
Observe
April 2002 TM MATH: Patterns & Growth 36
Observe
• C 264
• A 440
• E 330
• C 528
• 264/440 = 3/5
• 330/528 = 5/8
April 2002 TM MATH: Patterns & Growth 37
Note
144
89
89
55
April 2002 TM MATH: Patterns & Growth 38
April 2002 TM MATH: Patterns & Growth 39
April 2002 TM MATH: Patterns & Growth 40
April 2002 TM MATH: Patterns & Growth 41
April 2002 TM MATH: Patterns & Growth 42
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
April 2002 TM MATH: Patterns & Growth 43
References