They are formed as follow:

if n is even: divide by 2

if n is odd: multiply by 3 and add 1

 


For example, starting with 10 we get this sequence:

10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

we can see that this sequence loops into an infinitely repeating 4,2,1 sequence.  Trying another number, say 58:

58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

and we see the same loop of 4,2,1. These numbers are a kind of black hole numbers.


Hailstone numbers are called as such because they fall, reach one (the ground) before bouncing up again.  The proper mathematical name for this investigation is the Collatz conjecture. This was made in 1937 by a German mathematian, Lothar Collatz.

 

The following graphic from wikipedia shows how different numbers (x axis) take a different number of iterations (y axis) to reach 1. We can see that some numbers take much longer than others to reach one

 

 

References:

http://ibmathsresources.com/2015/01/30/hailstone-numbers/

http://mathworld.wolfram.com/HailstoneNumber.html/

https://en.wikipedia.org/wiki/Collatz_conjecture/

 

 

 

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