Patterns Fail

   
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A mathematics talk given by Colin Wright, this 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 ...

Careful ...

Some fail less quickly:

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


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?


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