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.
Discussion
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CategoryMaths