a proof that (N,0,N) is not solvable (always) because of the symmetry.

For every (N,0,N) instance (N truthtellers, 0 liars and N

randoms) there are a number of possible arrangements.

This number is always even, since for each arrangement there is a mirror arrangement where truth-tellers take the place of

randoms and viceversa.

And no arrangement is it's own mirror.

E.g. mirror(TRRT)=RTTR, mirror(TR)=RT.

Note that this mirror argument (the mirror of an arrangement is an arrangement of the same type of instance) only works for (N,0,N)-instances.

Assume that a successful interrogation strategy exists for the (N,0,N)-instance. Let M be maximum number of questions this strategy needs in the worst-case scenario. This means that in at most M number of questions, one can tell who the

randoms are, independent of the

random answerers' answers/strategy/

randomness.

Assume the following infinite-strategy: "I will act as if I am the truth-teller standing on the same position in the mirror-universe".

I'll call this type of person

**Infinite-Look-Alike**.

He is neither

Random (since he's predictable), he is neither Truth-teller.

One might ask if he is a perpetual liar, but this is not relevant to the proof. Actually he lies only about the arrangement and tells the truth otherwise i.e. a

**Perfect-Liar** - choose a lie and stick with it without contradicting known facts

Now, since

random answerers are truly

random, there's no certain way to distinguish them from an Infinite-Look-Alike

**in a finite number of questions**. A

random guy (by himself) can act like the Infinite-Look-Alike indefinitely.

In all the possibilities of M consecutive

random answers, for any number M, there is a possibility where all

random answerers act like Infinite-Look-Alikes (for the first M questions directed at them).

Which means there is a possibility that they act like Infinite-Look-Alikes for the entire successful interrogation strategy.

The problem is, that in that mirror universe, the mirror-truth tellers act like Infinite-Look-Alikes, and mirror-

Randoms act like truth-tellers.

In these circumstances, that ideal-interrogation strategy, applied in the mirror universe, gives the exact opposite result - failing.

So, (N,0,N) is not solvable.