## Most recent change of PatternsFail

Edit made on September 24, 2009 by DerekCouzens at 07:08:25

Deleted text in red / Inserted text in green

WW
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
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!! 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 ...
COLUMN_START
| /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 |
COLUMN_SPLIT
| /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 |
COLUMN_SPLIT
| /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 |
COLUMN_END

** It seems to be the primenumbers, prime numbers, but is it really?

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!! Some fail even more slowly
For each number, colour it black if it has an odd number
of primenumber prime number factors, and red if it has an even number of
primenumber 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

| 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?

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!! Some fail astonishingly slowly:
* /999*(n^2)+1/ is never a perfect square.
** Want a bet?

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!! Discussion

This is *your* page ...

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CategoryMaths