A puzzle on the Collatz conjecture

[Stouffer]

his conjecture has never been proven. However, it easily seen that, assuming positive integer n satisfies the conjecture, then 2n also satisfies the conjecture, since the path for 2n must start 2n, n and then follow the same path as n.


Beta Alanine

[Dirk Broer]

Collatz conjuncture (from wikipedia):
Take any natural number ''n''. If ''n'' is even, divide it by 2 to get ''n'' / 2, if ''n'' is odd multiply it by 3 and add 1 to obtain 3''n'' + 1. Repeat the process (which has been called "Half Or Triple Plus One", or '''HOTPO''') indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called '''oneness'''.

So, if you can start with any natural number "n", 2n will also work -and alsways be an even number as well-. As you keep on halving (or multiplying with 3 and adding 1 in case you reach an odd natural number) you will reach 1 in the end.

Say you start with natural number 38. 2n=76, 76/2=38, 38/2=19, (3*19)+1=58, 58/2=29, (29*3)+1=88, 88/2=44, 44/2=22, 22/2=11, (3*11)+1=34, 34/2=17, (3*17)+1=52, 52/2=26, 26/2=13, (3*13)+1=40, 40/2=20, 20/2=10, 10/2=5, (3*5)+1=16, 16/2=8, 8/2=4, 4/2=2, 2/2=1

You always reach 1 because you also will always reach one or more of the following natural numbers: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, etc.
Familiair numbers for those who double their computer memory from year to year, and have started some time ago.

[AMDave]

Illustrated and parodied
http://xkcd.com/710/

[Dirk Broer]

Illustrated and parodied
http://xkcd.com/710/

Hilarious!