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(n1)+k(n2)
 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