Knights and Liars of the Round Table (Posted on 2011-06-10)

The Knights and Liars of the Round Table is a group of knights and liars. Each person is either a knight or a liar, and at least one of them is a knight. The number of knights and liars in all is an even number greater than 20, but less than 30. One day, they were sitting around a round table. Each one of them said, "I am sitting between two people of different types." How many people are there in all, and how many are there of each type?

There is at least one knight (K_{1}). The two people next to this knight are of different types, so one is a knight and one is a liar. We will make the person on the left be the knight.

K_{2}K_{1}L_{1}

Now, the two people next to K_{2} are of different types, so there is a liar on the left of K_{2}. The two people next to L_{1} are of the same type, so there is a knight on the right of L_{1}.

L_{2}K_{2}K_{1}L_{1}K_{3}

The two people next to L_{2} are of the same type, so there is a knight on the left of L_{2}. The two people next to K_{3} are of different types, so there is a knight on the right of K_{3}.

K_{4}L_{2}K_{2}K_{1}L_{1}K_{3}K_{5}

We get the pattern ...KLKKLK..., where every third person is a liar. Therefore, the number of people must be divisible by 3. It is even, so it is divisible by 6. The only multiple of 6 that is greater than 20, but less than 30, is 24. Therefore, there are 24 Knights and Liars of the Round Table. Every third person is a liar, so there are 8 liars and 16 knights.

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