This presentation starts with
some seductively obvious patterns that seem successfully
to predict the future, but then goes on to show that not
all patterns are trustworthy.
It's all too common try try a few examples, find a pattern,
try a few more examples, see that the pattern continues,
and then leap to the conclusion that the pattern continues
forever.
Beware!
This talk gives some examples of patterns that look solid,
but which fail, often spectacularly. It goes on to explore
the notion of proof in mathematics, and why there are times
when we need to be certain.
Some patterns fail quickly:
Take a circle and put n points on its circumference.
Join them all with straight lines. How many pieces can you
get? The points don't need to be equally spaced - what's
the largest number of pieces?
| Points | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ... |
| Pieces | 1 | 2 | 4 | 8 | 16 | . | . | . | . | 256 | ... |
- What is the formula?
- How can you tell?
Careful ...
Some fail less quickly:
- k(1) = 0
- k(2) = 2
- k(3) = 3
- k(n+1) = k(n-1)+k(n-2)
- For what values of n does n divide k(n) ?
Here are the first few values ...
| n | Divides | k(n) |
|
1
|
Yes
|
0
|
|
2
|
Yes
|
2
|
|
3
|
Yes
|
3
|
|
4
|
No
|
2
|
|
5
|
Yes
|
5
|
|
6
|
No
|
5
|
|
7
|
Yes
|
7
|
|
8
|
No
|
10
|
|
9
|
No
|
12
|
|
10
|
No
|
17
|
|
11
|
Yes
|
22
|
|
12
|
No
|
29
|
|
| n | Divides | k(n) |
|
13
|
Yes
|
39
|
|
14
|
No
|
51
|
|
15
|
No
|
68
|
|
16
|
No
|
90
|
|
17
|
Yes
|
119
|
|
18
|
No
|
158
|
|
19
|
Yes
|
209
|
|
20
|
No
|
277
|
|
21
|
No
|
367
|
|
22
|
No
|
486
|
|
23
|
Yes
|
644
|
|
24
|
No
|
853
|
|
| n | Divides | k(n) |
|
25
|
No
|
1130
|
|
26
|
No
|
1497
|
|
27
|
No
|
1983
|
|
28
|
No
|
2627
|
|
29
|
Yes
|
3480
|
|
30
|
No
|
4610
|
|
31
|
Yes
|
6107
|
|
32
|
No
|
8090
|
|
33
|
No
|
10717
|
|
34
|
No
|
14197
|
|
35
|
No
|
18807
|
|
36
|
No
|
24914
|
|
- What do you think the answer will be for 37?
- It seems to be the prime numbers, but is it really?
Some fail astonishly slowly ...
For each number, colour it black if it has an odd number
of prime factors, and red if it has an even number of
prime factors. Count each prime factor each time it
appears, so 12 has an odd number of prime factors, 2, 2 and 3
Now start from 2 and count +1 for each black number and -1
for each red number. It seems that the blacks are always
ahead.
| Number | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | ... |
| Factors | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 2 | 2 | 1 | 3 | 1 | 2 | 2 | 4 | 1 | 3 | 1 | 3 | ... |
| "Sign" | + | + | - | + | - | + | + | - | - | + | + | + | - | - | - | + | + | + | + | ... |
| Sum | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 2 | 1 | 2 | 3 | 4 | 3 | 2 | 1 | 2 | 3 | 4 | 5 | ... |
Are they always?
So when can you trust a pattern?
Contents
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Colin and
Rachel Wright:
- Maths, Design, Juggling, Computing,
- Embroidery, Proof-reading,
- and other clever stuff.
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