# Patterns Fail

Sometimes patterns do go on forever, but sometimes apparent patterns fail.

## Some fail quickly:

• Slice the cake with straight cuts between points on the perimeter.
• One point gives 1 piece
• Two points gives 2 pieces
• Three points gives 4 pieces
• Four points gives 8 pieces
• Five points gives 16 pieces
• Clearly six points should give ??
See SlicingTheCake

## 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 k(n) Divides 1 0 Yes 2 2 Yes 3 3 Yes 4 2 No 5 5 Yes 6 5 No 7 7 Yes 8 10 No
 n k(n) Divides 9 12 No 10 17 No 11 22 Yes 12 29 No 13 39 Yes 14 51 No 15 68 No 16 90 No
 n k(n) Divides 17 119 Yes 18 158 No 19 209 Yes 20 277 No 21 367 No 22 486 No 23 644 Yes 24 853 No

## Some fail even more slowly

For each number, colour it black if it has an odd number of prime number factors, and red if it has an even number of prime number factors. Count each 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?

## Some fail astonishingly slowly:

• 999*(n^2)+1 is never a perfect square.
• Want a bet?