For a given nonzero natural integer *n*=*n*_{0},
one constructs a list of
integers *n*_{1}, *n*_{2}, *n*_{3},... in
the following way. If *n*_{0} is even
then *n*_{1}=*n*_{0}/2. If *n*_{0}
is odd then *n*_{1}=3*n*_{0}+1. The next
element, *n*_{2}, of the list of integers is determined
by the same rule applied to *n*_{1}:
if *n*_{1} is even
then *n*_{2}=*n*_{1}/2. If *n*_{1}
is odd then *n*_{2}=3*n*_{1}+1.

For example, if *n*_{0}=1
then *n*_{1}=4, *n*_{2}=2, *n*_{3}=1, *n*_{4}4,
etc. We have a cycle: 1,4,2,1,4,2,1,4,2,1,...

Let us start with another value
for *n*_{0}. If *n*_{0}=3, we find the list
10,5,16,8,4,2,1,4,2,1,4,2,1,...

Let us try yet another one, starting with 7: 22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,...

It seems that we eventually fall into the cycle 1,4,2,1,... This is the Syracuse Problem, do we always end up in the cycle 1,4,2,1,.. whatever initial value we take? In fact, often the problem is posed by asking whether we arrive eventually at 1 starting with any nonzero natural integer.

It is interesting to write the integers of the list in binary representation. Graphically, a black disc will be a 1 and a white disc will be a 0 digit. Drawing the numbers in a spiral, least significant bits in the interior, most significant bits at the rim, we get the following kind of dynamics:

The starting value here is 18446744073709551617
which happens to be equal to 2^{26}+1. This explains why at the start we only have two black discs. the movie stops when we reach 1. If one wants to control the movie, one can install gif reader and activate it on this page.

Here is another one.