Why aren't the numbers playing nicely anymore?

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

• Love the brackets! May 28 '15 at 19:44
• Some serious observations: almost all in the first group are composite; all in the third group end in 1, 3, or 7; and are you sure this isn't a duplicate? May 28 '15 at 19:50
• @randal'thor I couldn't find a puzzle posted with this solution but it's hard to find number sequence puzzles. If you find a duplicate, I'll vote to close this myself. May 28 '15 at 20:00
• @Vicky I opted to define natural numbers as positive integers per OEIS A000027. Zero is sometimes included in natural numbers and sometimes not. Given the rule by which the groups are divided, zero would either be in the first group or off sulking in the corner because it's not sure where it belongs. May 29 '15 at 16:18
• if I wanted to know which group arbitrary X is in, would I need to work out the positions of the numbers up to X, or can I apply the rule to X "standalone"? Jun 3 '15 at 12:39

After some analysis, it appears the numbers are suffering from some common psychological problems:

The first group:

Has an inferiority complex. They stick together because when spelled out, their names have fewer letters than the number that came before them.

The second group:

Is obsessive compulsive. They stick together because they have exactly the same number of letters as the number that came before them.

The third group:

Has a superiority complex. They stick together because their names have more letters than the number that came before them.

Treatment is tough; the only language that comes to mind that "fixes" this problem is Chinese, as some basic googling seems to show that each number is represented by only a single character up to a certain point. Even then, however, I'm sure the numbers would start squabbling about how many strokes it takes to form the character!

• Congratulations, Herr Doctor! You have figured out the grouping and even the psychological root cause. Your solution is a reasonable one. The only option I have devised is to switch to a unary numeral system but that's essentially destroying 90% of the digits. They numbers can get along so long as there's only one in existence but that's a bit drastic. Jun 3 '15 at 15:11
• Another idea might be to work with the numbers to have them think of themselves as numerals rather than words. This should be easy, considering words are collections of letters, and numbers are notoriously distrustful of letters. Once that's accomplished, convince them that all numbers have an infinite number of leading zeroes, so all numbers have the same number of digits. Of course, at that point the Integerverse opens a portal to the Naturalverse and the negative numbers pour in to lord their superiority over their "signless" brethren... Jun 3 '15 at 15:20
• Nice one! God, I hate superiority complexes :-/ Jun 3 '15 at 19:05
• @randal'thor Ugh, I know, people who are full of themselves are so beneath me! Jun 4 '15 at 1:41

With the sole exceptions of $1, 14, 15, 16, 18, 19,$ the numbers up to $40$ are divided as follows:

all those ending in 4,6,9,0 in the first group; all those ending in 2,5,8 in the second group; all those ending in 1,3,7 in the third group.

While each of these exceptions is

in the same group as double itself ($14$ with $28$, $15$ with $30$, and so on) - except for $1$, which is in the $1$st group (makes sense: that $1$ is so $1$-y, you know).

So my answer to the official question is:

the numbers divided themselves up according to their congruence class modulo 10 (i.e. final digit), except for the number $1$, who had to be in the $1$st group to preserve $1$-iness, and the teenagers, who - just to be contrary - joined a group according to the congruence class of double their value instead of themselves. The latter could be explained either by those teenagers being so full of themselves they think they're worth twice as much as they are, or by interpreting $2n$ as the mother of $n$ and saying they want to stick close to their mothers.

And my answer to the bonus question:

tell all the teenagers' mothers to type "problem teenager" into Google, absorb all the wisdom of the internet (Parenting Stack Exchange, perhaps?), educate their teenagers in how not to squabble (it was those teenagers who started the whole argument anyway), and then the rift between the adult numbers will automatically be healed too.

OK, so I'm a rubbish number therapist when it comes to the bonus question. But at least the answer to the official question works and is in the spirit of the OP!

• I'm pretty sure if you need to express it like that, it's not right. Also, given that it's "1, 14, 15, 16, 18, and 19", that's a strong red flag for English representations.
– user88
May 29 '15 at 0:05
• You have achieved an 85% solution. However, all the numbers followed the same, singular rule. Please continue striving for the 100% solution. May 29 '15 at 1:43

Partial answer and I am unsure how to hide the answer but for the 3rd group of numbers the pattern seems pretty obvious of

the sequence of differences between the numbers is 4 4 2 4 4 2 4 4 2.

• This pattern - along with the similar patterns found at the end of the other two groups - is an artifact of the underlying rule. It may or may not help you find that rule. Every number is sorted by a single rule, though. Jun 1 '15 at 16:31