Patterns and Growth

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Patterns and Growth

• John Hutchinson

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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?

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People Handshakes
1 0
2 1
3
4
5
6
7
4
People Handshakes
1 0
2 1
3 3
4
5
6
7
5
People Handshakes
1 0
2 1
3 3
4 6
5
6
7
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People Handshakes
1 0
2 1
3 3
4 6
5 10
6 15
7 21
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Is there a pattern?
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Heres 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
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Heres 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
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What is
1 2 3 4 .. 98 99 100?
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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 (100101)/2 5050.
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Look at
1 2 3 4 5 6 3 ? 7 21
0 1 2 3 4 5 6 7 4 ? 7 28
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If there are n people in a room the number of
handshakes is
n(n-1)/2.
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Problem 2 How many intersections?

• Given several straight lines. In how many ways
  can they intersect?

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2 Lines
1
0
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3 Lines
0 intersections
1 intersection
2 intersections
3 intersections
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Problem 2A

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

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Is the pattern familiar?
Lines Intersections
1 0
2 1
3 3
4 6
5 10
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Problem 2B

• Up to the maximum, are all intersections possible?

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What about four lines?
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What about two intersections?
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What about two intersections?
Need three dimensions.
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Problem 3

• What is the pattern?

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

• 1 1 2
• 1 2 3
• 2 3 5
• 3 5 8
• 5 8 13
• 8 13 21
• 13 21 43

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This is the Fibonacci Sequence.
Fn2 Fn1 Fn
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Divisibility

• Every 3rd Fibonacci number is divisible by 2.
• Every 4th Fibonacci number is divisible by 3.
• Every 5th Fibonacci number is divisible by 5.
• Every 6th Fibonacci number is divisible by 8.
• Every 7th Fibonacci number is divisible by 13.
• Every 8th Fibonacci number is divisible by 21.

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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
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Pascals Triangle
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1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
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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
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Note
1 1 2 3 5
8
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
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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?

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Lets 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
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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?

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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
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Observe
5
2
3
C
D
E
F
G
A
B
C
8
13
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Observe

• C ? 264
• A ? 440
• E ? 330
• C ? 528

• 264/440 3/5
• 330/528 5/8

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Note
89
55
144
89
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Flowers
Petals Flower Flower Flower
1 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
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References