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There must be some Liars at the table, as the knights would not otherwise claim that they were sitting next to Liars.
For a Liar to state that he was sitting between a Knight and another Liar, he must actually be sitting either between two Liars or two Knights.
But if we have two Liars next to one another, we can have no Knights at the table, since if we have a K-L-L combination, the Liar in the middle would be sitting between a Knight and a Liar, and thus would have told the truth.
If everyone at the table were Liars, they could say that they sat between a Liar and a Knight and still be true to form, but the problem states that there were at least two Knights at the table, so we must conclude that no two Liars were sitting next to one another.
In fact the only feasible setup seems to be a repeated sequence of two Knights followed by one Liar:
K-K-L-K-K-L-K-K-L ....
This sequence repeated 15 times would yield all liars sitting between two Knights, and each Knight being between a Knight and a Liar.
Note that since the total number of people - 45 - can be divided evenly by 3, the last person at the table - a Liar, would be sitting between Knight #44 and Knight #1:
042 043 044 045 001 002
L K K L K K
But the problem also states that two of the Knights were mistaken, and therefore each of the two was actually either sitting between two other Knights or two Liars.
If both were sitting between Liars, that means two of the "K-K-L" sequences turned into "K-L". But then the order would be "shortened" by two people, which would create a "mismatch" at the end of the table:
042 043 044 045 001 002
K L K K K K
Creating a situation where two extra Knights are sitting between two other Knights, and are also mistaken. (So we would get a total of 4 mistaken knights instead of two.)
The same problem occurs if we try to seat both errant Knights between two Knights each.
The only possible solution, is to have one of the mistaken Knights sit between two Liars, and the other - between two Knights. The result - the balance of Knights to Liars is unchanged - of the 45 guests, 30 are Knights, and 15 are Liars. |
