The game, which is purely recreational, goes as follows:
Starting out with 1, you can employ any of two different generation rules:
- You can multiply by 3
- You can divide by two, rounding up (e.g. 3 divided by 2 is 2)
Solving the game, i.e. showing that you can reach every number is another interesting problem, although trivial in comparison to the property I am trying to define: complexity. Say the complexity of an integer N is the shortest sequence of rules needed to reach N.
As an example, the complexity of 9 is 2. The derivation has the following notation:
1,3,9.
So we write C(9)=2. There are some numbers, namely all powers of three, which have a low complexity. But even small numbers such as 4 or 10 are already very complex.
I need your help in researching the complexity of a number and its inherent properties (lower, upper bounds). The randomness seems to be insane, mostly stemming from the "round-up" rule, which yields unpredictable results.
I leave you with some derivations so that you can get a feel for how you reach a number:
(7): 1,3,9,27,14,7. (5): 1,3,9,5. (11): 1,3,9,27,14,7,21,11. ...´
I will be monitoring the comments for additional information requests.