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).

  • $\begingroup$ 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. $\endgroup$
    – Sconibulus
    Sep 26, 2016 at 17:36
  • 1
    $\begingroup$ 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 $\endgroup$
    – Joe
    Sep 26, 2016 at 18:19
  • $\begingroup$ @Joe, interesting idea. thanks. It is hard to group medals, though. $\endgroup$
    – klm123
    Sep 26, 2016 at 18:27
  • $\begingroup$ 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. $\endgroup$ Sep 26, 2016 at 18:30

1 Answer 1


A simplistic solution would be to describe the 3 people rather than counting them - there is a village where every household is inhabited by a single family. Each family has either one daughter or have one son; none have both.

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.

This could also introduce some fluff intended to confuse the solver as to whether a child being a son or a daughter is relevant to the puzzle.

  • 1
    $\begingroup$ Yeah. But the problem with concept of family is that it is complex one and the solver will start to analyze it imidiately. As result all he needs to do is to add up 2+1. Additionally he will as questions, like do grantparrents belong to a family? $\endgroup$
    – klm123
    Sep 26, 2016 at 15:24
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    $\begingroup$ Romantic triangle is kind of similar. $\endgroup$
    – Matsmath
    Sep 26, 2016 at 16:22
  • $\begingroup$ One problem I see here is that the number of relationships is constrained. While towns can have any number of highways, a father will always have one wife and one child, a mother will always have one husband and one child, and how many parents can a child have at most? The romantic triangle seems more flexible... $\endgroup$
    – GOTO 0
    Sep 26, 2016 at 17:50
  • $\begingroup$ @Matsmath, Romantic TRIangle doesn't work for obvious reasons:) Squares could do better... $\endgroup$
    – klm123
    Sep 26, 2016 at 18:07

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