4. Around a circular table, 6 people are seated, each of whom can either be a knight (who always tells the truth) or a knave (who always lies). Each of those present states: "Considering my two neighbors and the person sitting directly opposite me, exactly two of these three people are knaves." How many knights are there in total at the table?
(A) none
$\begin{array}{ll}\text { (B) } 1 & \text { (C) } 3\end{array}$
(D) 4
(E) the information provided is not sufficient to determine it
(4) The correct answer is $(A)$.

If there is at least one knight, among his two neighbors and the person opposite, there are exactly two liars. Since there are at least two liars, and at most four if the other two people are also liars, the knights are two, three, or four. Moreover, each knight has exactly one neighboring knight or has a knight sitting directly opposite, but not both. If the knights are exactly two, they must be sitting next to each other or opposite each other. But if they are sitting next to each other, a liar sitting in the opposite position tells the truth; if they are sitting opposite each other, a liar next to them tells the truth. We can conclude that the knights are not exactly two. If the knights are exactly three, two of them must necessarily be sitting next to each other, but not all three, otherwise a knight would have two neighboring knights. The third knight must be opposite one of the two, but then one of the first two has both a neighboring knight and one opposite. We can therefore conclude that the knights are not exactly three. If the knights are exactly four, at least two are sitting next to each other. But each of them can have at most one neighboring knight, and therefore the other two must necessarily be sitting opposite. But then each knight has both a neighboring knight and one opposite, and we can conclude that the number of knights is not four either.

The only other possibility is that there are no knights, and only liars are sitting at the table. In this case, everyone lies and the situation is possible.

Problem proposed by Matteo Protopapa.