In the land of Zoz, there are three types of people. In addition to the usual Knights and Liars, there are Switchkins who become whatever they say they are.

One morning, three groups of 30 gather. The first group has one type, the second group has an equal number of two types, and the third group has an equal number of all three types.

Everybody in one group says "We are all Knights", everybody in another group says "We are all Liars", and everybody in the remaining group says "We are all Switchkins."

How many Liars are there after this announcement?

Obviously, the group which says 'we are all liars' is all switchkins. If there were any knights, they would be lying, and if there were any liars, they would be telling the truth. That makes 30 liars so far.

Liars: 30

If then follows that there cannot be any knights in the group which says they are all switchkins, as they would be lying. This group must then be the group composed of two types - liars and switchkins. This adds 15 liars.

Liars: 45

Then, the last group, which says that they are all knights, must be composed of ten liars, ten switchkins, and ten knights. This adds ten liars.

Liars: 55