The natural numbers have gotten in a big fight and split up into groups. If we can't get them back together again, the universe may never be the same! What kind of sense does it make to have three counting systems? No, they must be reunited.
You are a licensed number therapist. Your first step is to figure out what the divisive issue was but none of the numbers are talking. Your only choice is to deduce the rule under which they split into groups based solely on which numbers are in which group. Once you get that far, you're pretty sure you can get them all back together again.
These are the three groups up to the number $40$:
\begin{array}{c} 1\ 4\ 6\ 9\ 10\ 15\ 18\ 20\ 24\ 26\ 29\ 30\ 34\ 36\ 39\ 40\ \ldots \\ \hline 2\ 5\ 8\ 12\ 14\ 16\ 19\ 22\ 25\ 28\ 32\ 35\ 38\ \ldots \\ \hline 3\ 7\ 11\ 13\ 17\ 21\ 23\ 27\ 31\ 33\ 37\ \ldots \end{array}
Every natural number fits into one of the three groups (You can see the groups stretching into infinity (You have very good eyes (It's practically a necessity for a number therapist.).).).
OFFICIAL QUESTION: What rule determines which group each number joins?
BONUS OFF-TOPIC QUESTION: What sort of therapy session would you design to bring the numbers together again? (Not to be submitted without a serious attempt at the official question.)
Hint:
There was a second tag added on Friday
Hint:
The rule can only be found when you spell out the cardinal name of each number in English