In every group of shepherds in a certain island, at least one is a knight, who always speaks truthfully, and at least one is a liar, who always speaks falsely.

A visitor approached four shepherds (denoted by A, B, C and D) on a hillside and asked each how many of the four were knights. These answers were given:

A: Three of us are knights.
B: One of us is a knight.
C: Two of us are knights.
D: None of us is a knight.

The visitor approached four more shepherds on another hillside (denoted by E, F, G and H) and asked how many were liars. Their answers follow:

E: We are all liars.
F: One of us is a liar.
G: Three of us are liars.
Shepherd H declined to speak.

How many of the shepherds on each of the two hillsides were knights?

The first group is pretty straight forward.

D has to be a Liar, as it is a given that at least 1 of them is a knight.

Of the remaining 3, since all are making contradictory statements, only 1 can possibly be a Knight.

Hence, B is a Knight and A C and D are Liars.

The other group starts in a similar manner, but gets tricky with subsequent statements ... more so due to the hush of a certain Mr H.

E has to be a Liar, as it is a given that at least 1 of them is a Knight.

Between F and G, both cannot be knights, due to their contradictory statements.

If both of F and G were Liars, then H has to be the Knight ... but then G's statement turns out to be true.

So between F and G ... one is a knight and the other is a Liar ... irrespective of what H is!<o:p></o:p>

F cannot be a knight, as then G has to be a knight as well ... [we have already determined that E has to be a Liar]<o:p></o:p>

So F is a Liar, which makes G a knight, and H a liar ... despite his silence!

Posted by Syzygy on 2012-07-11 02:28:37