A number of Knights and Liars went on a camping trip. Having pitched their tents for the night at the end of a long day's hike, Thomas (the best cook by far) settled down near the camp fire to make stew whilst everyone else sat in a circle around him, watching. Looking around, Thomas noticed that each person seemed to be sat between two people they knew, whereas Thomas himself knew no-one except his good friend Richard. So, getting everyone's attention, he asked a person at random in the circle the following question:

"You and the two people that are sitting next to you: Is there an odd number of Liars in that little group?"

The person replied. Thomas asked another person at random, and that person gave the same reply as the first. Again and again he asked and every time the reply was the same. Finally, having asked everyone else and always receiving the same reply, he turned to Richard and asked the question once more. Surprisingly, Richard answered differently to everyone else.

Thinking for a moment, Thomas asked Richard: "Are you sitting between two Knights?", to which Richard smiled and gave the same reply as he had previously.

Nodding, Thomas declared: "So, the Knights are outnumbered by the Liars here!", and turned back to making the stew once more.

If "n" people in total went on the camping trip, how many Knights and Liars are there, and what are Thomas and Richard?

(In reply to What is truth? (Part 1) by TomM)

If Richard's answer is "yes," then the group alternates Knight and Liar, starting at the person on Richard's left, and proceeding clockwise to the person on Richard's right. Each of them believes Richard to be of the opposite temperment from themselves, but Richard believes one of them to be the same. Since everyone is sitting between two friends, they aren't mistaken in their beliefs, so this is a contradiction. Richard can not have answered "yes".

So Richard answered "no." Going around the circle, starting with the person on Richard's left, you have 2 Knights or 2 Liars, followed by 2 of the opposite, and so on, until you get to the last pair before Richard, who are the same temperment as the first pair. Richard is the opposite temperment. Either way, he had to answer "yes" to the second question. (Perhaps that's why he smiled before he answered. The second question adds no new information.)

Posted by TomM on 2002-06-23 10:44:54