Mathematical mysteries: Hailstone sequences
Issue 1January 1997
This problem is easy to describe but it is one of mathematics' unsolved problems.
Starting with any positive integer n, form a sequence in the following way:
- If n is even, divide it by 2 to give n' = n/2.
- If n is odd, multiply it by 3 and add 1 to give n' = 3n + 1.
Then take n' as the new starting number and repeat the process. For example:
- n = 5 gives the sequence
- 5, 16, 8, 4, 2, 1, 4, 2, 1,...
- n = 11 gives the sequence
- 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...
These are sometimes called "Hailstone sequences" because they go up and down just like a hailstone in a cloud before crashing to Earth - the endless cycle 4, 2, 1, 4, 2, 1. It seems from experiment that such a sequence will always eventually end in this repeating cycle 4, 2, 1, 4, 2, 1,... and so on, but some values for N generate many values before the repeating cycle begins. For example, try starting with n = 27. See if you can find starting values that generate even longer sequences.
An unsolved problem is, can it be proved that every starting value will generate a sequence that eventually settles to 4, 2, 1, 4, 2, 1,...? Could there be a sequence that never settles down to a repeating cycle at all?
Hailstone Evaluator
Enter any positive integer, the Hailstone sequence will be returned.
Comments
the mother function of 3n+1
there i s function that when you let n=1 the collatz problem appears. but also, when N=-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8...,infinity. as you guys can see, if we solve 3n +1, we will solve all of them. with the function is easy to find and loo of every roblem. why all numbers top at 1, is just because there are not positive numbers less than 0.
Something to think about...
It would be very interesting to know whether or not "stopping length" as a function of n exhibited any regularity, and if so, of what nature? If you plotted the "stopping length" of each sequence against the unique n that generated it, what would the graph look like?
Maybe someone with some programming ability can try this out...
There is an obvious pattern
Try 55, 115, 175, 235. All these have something in common
List of Lengths
My lengths are defined by stopping once a 4 is reached.
n=5 len=3
n=6 len=6
n=7 len=14
n=9 len=17
n=18 len=18
n=25 len=21
n=27 len=109
n=54 len=110
n=73 len=113
n=97 len=116
n=129 len=119
n=171 len=122
n=231 len=125
n=313 len=128
n=327 len=141
n=649 len=142
n=703 len=168
n=871 len=176
n=1161 len=179
n=2223 len=180
n=2463 len=206
n=2919 len=214
n=3711 len=235
n=6171 len=259
n=10971 len=265
n=13255 len=273
n=17647 len=276
n=23529 len=279
n=26623 len=305
n=34239 len=308
n=35655 len=321
n=52527 len=337
n=77031 len=348
n=106239 len=351
n=142587 len=372
n=156159 len=380
n=216367 len=383
n=230631 len=440
n=410011 len=446
n=511935 len=467
n=626331 len=506
n=837799 len=522
n=1117065 len=525
n=1501353 len=528
n=1723519 len=554
n=2298025 len=557
n=3064033 len=560
n=3542887 len=581
n=3732423 len=594
n=5649499 len=610
n=6649279 len=662
n=8400511 len=683
n=11200681 len=686
n=14934241 len=689
n=15733191 len=702
n=31466382 len=703
n=36791535 len=742
n=63728127 len=947
n=127456254 len=948
n=169941673 len=951
n=226588897 len=954
n=268549803 len=962
n=537099606 len=963
n=670617279 len=984
code error
i tested you program and it failed for number 5
the sequence for 5 is 5, 16, 8, 4, 2, 1 definitly not 5, 16, 8, 4, 2, 1, 4, 2, 1,...
code error
when you get to 1 (which is odd) the next result is 4, 2, 1....
I think you missed the point
Try reading the article next time...
It seems from experiment that such a sequence will always eventually end in this repeating cycle 4, 2, 1, 4, 2, 1,...
Collatz
This is also called the Collatz conjecture. This is the name of the unsolved problem. Perhaps you should mention this in the article