What is an example of a concept with implicit trio-relationship?

I'm trying to figure out a puzzle and here is my problem.

Let's say we want to create a puzzle basing on the idea of distinguishing odd and even numbers. It can be something like this:

There is an# country with exactly 123 towns, towns are connected with highways. Is it possible, that the 1st town has one highway coming out of it, 2nd - 2, ..., 123th - 123?

No. Let's suppose that it is possible. Then we add up all ends of highways. We will get (1+2+...+123) - an odd number of ends, but it always should be even!

So here we have relations between towns, which are implicitly dual. We never needed to say that a road have exactly 2 ends, because it is common knowledge.

Now imagine we want to create a puzzle where the solver needs to take not "mod 2", but "mod 3". We can do it like this:

There is a village with exactly 1234 people. The people are members of clubs, and each club have exactly 3 members. Is it possible, that the 1st guy belongs to 1 club, 2nd - to 2, ..., 1234th - to 1234?

But the problem is the big shinny number 3 in the condition itself. It is needed to make trio-relations, but it is a huge hint. So my question is how to rid of it and make trio-relations implicit?

I just can't figure out a concept, which is common knowledge, and which will always put 3 objects together, like a concept of "line" puts 2 ends together. (quad- or penta-relations would do too).

• Several games have 3v3 formats; perhaps these could be Magic: The Gathering Emperor teams, or League of Legends: Twisted Treeline teams. These might let you hide a clue in something that looks like flavor text. Sep 26, 2016 at 17:36
• Maybe you could do something with medals - gold/silver/bronze are so strongly implied by just saying "medals" you probably don't need to go into too much depth. The main trouble would be finding a casual explanation for it. I've been chewing it over but can't think of something concise yet
– Joe
Sep 26, 2016 at 18:19
• @Joe, interesting idea. thanks. It is hard to group medals, though. Sep 26, 2016 at 18:27
• BTW, why does a road need to have exactly 2 ends? Can't there be a "fork" in the road (you know - the kind where truth-tellers and liars loiter about in wait for a passing logician)? Also, thousands of roads interconnecting 123 towns are bound to cross each other several times. Sep 26, 2016 at 18:30

Since a family has to contain a mother and a father (assuming we're not tagging the question as lateral-thinking), the one child will make 3 people but we've not actually counted them explicitly.