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?
the answer is
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).