Whose Group? (Outliers of Hyperborea Part I)

Hyperboreans belong to three groups: Sororeans, Nororeans, and Midroreans. Of the three inhabitants who make the statements below, little is known as to their group or groups except that Outliers are not present.

1. 1. B and I are both Midroreans.
   2. C is a Nororean.
2. 1. A and I belong to two different groups.
   2. C and I belong to the same group.
3. 1. I am not a Nororean.
   2. A and I belong to the same group.

To what group or groups do A, B, and C belong?

(Source: Challenging False Logic Puzzles by Norman D. Willis)

To start with, at least, I like to take the first person at their first word in these sorts of puzzles. A has claimed to be a Midrorean. This means either that A1 is true and A2 false, or that A1 is false and A2 true. Consider the first possibility. It means that B is a Midrorean. B's only possibility as a Midrorean then is constrained to B1 being false and B2 being true, which means that C is also a Midrorean. If C is a Midrorean then his only possibility are is C1 being true and C2 being false. However C2 is not false under the other assumptions and so the other possibility for A to be a Midrorean has to be considered.

If A1 is false and A2 is true then B isn't a Midrorean, which isn't too helpful because two possibilities remains for him. What is more helpful is that this then entails that C is a Nororean. If C is a Nororean, then C1 is false, which checks out, and that C2 is false, which also checks out. So far, it looks like the belief that A is a Midrorean can be preserved with the implication that C is a Nororean.

But not so fast. This leaves B. B can't be a Sororean, because C2 is false under this assumption. Nor can he be a Nororean because C1 is true under this assumption. All that remains is for B to be a Midrorean. In such case, because B1 is false, B2 must be true. But it is in fact false with this assumption. This requires revision of the belief that C is a Nororean and, indeed, of the belief that led to it, that A is a Midrorean at all. Disappointing, but it doesn't leave us quite at square one, because that latter possibility is now ruled out entirely.

What other possibility remains for A? Sororean is not in the cards; a Sororean would not identify as anything other than a Sororean. All that leaves is Nororean, and now we can be certain, although there are unfortunately no unique implications from the falsity of A1 and A2.

That being said, A being a Nororean does mean, due to the falsity of A2, that C is either a Sororean or a Midrorean. Entertain the first option in the spirit of charity. C2 is false under this assumption. So much for charity. This then means that C is a Midrorean, with C1 as the true statement and C2 as the false one. We can now be certain that A is a Nororean and C is a Midrorean. These aren't defeasible inferences anymore.

That leaves B. Being a Sororean is still ruled out because of B2. What about being a Midrorean? B1 must then be true, and B2 false, but this is not the case. All that remains is for B to be a Nororean.

Final answer: A is a Nororean, B is a Nororean, and C is a Midrorean.