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?

I will assume that not everybody in one group says "We are all Knights," as a Knight would not claim himself as a Liar or Switchkin, and at least one group has a Knight which also includes Liars and Switchkins. Rather, I will assume that each person within that group says "I am a Knight."  This group of all three types, with the Liars lying and the Switchkins choosing to become Knights, would then be composed of 10 Liars and 20 Knights -- 10 of which were Switchkins who chose to become Knights.

The group that says that they are all Liars must be all Switchkins.  The Knights would not lie, and the Liars would not tell the truth.  These Switchkins, of course, are now considered Liars, therefore 30 Liars are now in this group.

The group that says that they are all Switchkins, therefore, must be the group of two types.  The Switchkins can be telling the truth and the Liars, of course, can be lying.   There are 15 Liars and 15 Switchkins in this group.

The total Liars, therefore, to include the Switchkins that have chosen to become Liars, is 55.

Edited on July 18, 2006, 7:37 pm

Posted by Dej Mar on 2006-07-17 10:35:43