# Patterns Fail

 A Mathematics Talk given by Colin Wright Tagged As Talk Description
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?