There is a famous puzzle that asks for the next line (and the rule) for the following pattern:
The solution to this puzzle is:
312211, because each line describes the line before it. The first line has one '1', and so the second line is 11 (one '1'). The second line has two '1's (21) and so the third line is 21.
Now, the question is:
Is there a starting point, which would come back to itself? In other words, can you find a starting point and then run this rule over and over again and come back to the starting point (so it forms a closed cycle)?
Some clarifications. Usually, the numbers (after the first) will have an even number of digits. But not always. For example,
There is at least one solution to this puzzle. But are there more?