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?

First, if there were all Knights or all Liars in the circle, then everyone, including Richard, would have answered "no" to Thomas' question. If Richard were the only Knight or the only Liar, then he would have answered "yes," but so would have his two neighbors. So there are mixed Knights and Liars.

There are two possibilities: if the general answer was "yes," every Knight is between a Knight and a Liar and every Liar is between a Knight and a Liar. If the general answer was "no," every Knight is between two Liars, and every Liar is between two Knights. Either way, there are as many Knights as Liars, with some kind of exception in Richard's group.

Posted by TomM on 2002-06-23 10:43:56