In Tripleland, natives always go in trios: a knight, a knave, and a liar.

Once I met such a trio, and I asked one of the natives a simple question ("simple" meaning, "of six words or less"); he answered, and I knew what type he was. Then, I asked another of the natives a different simple question; he answered, and I knew what type he was, and therefore, the type of the third one too.

"Logical" thinking: This cannot be. The natives could be in six possible orders. Two yes-no questions allow four combinations. Thus, you cannot pick one out of six with only two questions; you need one more!

How could this be? What's wrong with the reasoning above?

There are two things wrong with the "logical" thinking.

First, the original paragraph didn't say anything about the questions being yes-no; they could, for example, be things like "what type are you?" or "what type is the previous speaker?" or "what type is the redhead here?".  Since there are three types, there are nine possible pairs of answers.

But, more significantly, as there are eight possibilities for three y/n answers and only six possible orders, two possibilities are redundant, and so allow for earlier solution in some cases.  A possible set of answer/identification pairings would be:

Y Y Y   Liar Knight Knave
Y Y N   Liar Knight Knave
Y N Y   Liar Knave Knight
Y N N   Liar Knave Knight
N Y Y   Knave Knight Liar
N Y N   Knave Liar Knight
N N Y   Knight Knave Liar
N N N   Knight Liar Knave

In the first four lines above, the first two questions have already answered the question of the order of the persons--the third question is redundant, in those cases only.  If the first two questions had been answered NY or NN, then, and only then, would the third question be necessary.

Edited on September 2, 2006, 11:54 am

Posted by Charlie on 2006-09-02 11:52:38