Edward and Edwin

My National Review Online "Diary" column for September 2003 included a brainteaser:  "Edward and Edwin."  I took this from Raymond Smullyan's engrossing little book of logic puzzles, The Riddle of Scheherazade.  Here is the puzzle, followed by a worked solution, both in Smullyan's precise words.

Edward and Edwin

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There is a pair of identical twins named Edward and Edwin, who are indistinguishable in appearance. One day shortly after they were grown, a strange disease struck them both and changed their lives forever. Henceforth, each twin was in one of three psychological states — State 1, or State 2, or State 3 — that alternated in a constant cyclical pattern: 1, 2, 3, 1, 2, 3, 1,... and so on. Curiously enough, at any given time, both brothers were in the same state — both were in either State 1, or State 2, or State 3. There was, however, a crucial difference. Edward always lied when he was in State 1, but told the truth in the other two states. Edwin, on the other hand, lied when in State 2, but told the truth when in State 1 or State 3.

One day, one of the brothers was asked: “Are you either Edwin in State 2 or Edward not in State 1?” From his answer, is it possible to deduce what state he is in? From his answer, can one deduce whether he is Edward or Edwin?

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Solution:

You cannot tell what state he is in, but you can tell who he is.

Suppose he answers yes.  If he is in a truthful state, then he really is either Edwin in State 2 or Edward not in State 1.  But he then can't be Edwin in State 2 (in which he lies); hence he must be Edward, but not in State 1.

On the other hand, if he lied, then, contrary to what he said, he is neither Edwin in State 2 nor Edward not in State 1; hence he is either Edwin not in State 2 (and thus in a truthful state) or Edwin in State 1, but he can't be Edwin not in State 2, since he lied; hence he must be Edward in State 1.

This proves that if he answers yes, he must be Edward (maybe in State 1 or maybe not).

Now, suppose he answers no.  If his answer is truthful, then he is neither Edwin in State 2 nor Edward not in State 1; hence he is either Edwin not in State 2 or Edward in State 1.  But he can't be Edward in State 1, since he told the truth; so he must be Edwin (but not in State 2).

On the other hand, if he lied, then he is either Edwin in State 2 or Edward not in State 1, but the latter alternative is not possible (since Edward not in State 1 doesn't lie), so he must then be Edwin in State 2.  Thus, if he answers no, he must be Edwin.

In summary, if he answers yes, he is Edward, and if he answers no, he is Edwin.