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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

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!! Some fail even more slowly

For each number, colour it black if it has an odd number

of

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?

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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