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?
One group which consists of only one type of people must be a group purely of liars and they say "we are all switchkins". This is because
1) knights only speak the truth and hence they can be present only in the group which says "we are all knights".
2)If this group which says that "we are all knights" is the one which consists only of one type of people then the other two groups are one which says "we all are liars" and one which says "we all are switchkins" and knights must be present in atleast one of these two group which contain equal no of all three types of people. But this is not possible as knights never lie.
3) Therefore the group which consists of knights and which says "we are all knights" cannot be a group purely of knights. It should consists of either knights and liars/switchkins in equal number or knights, liars and switchkins in equal number. Aagin, it cannot be a group containing equal number of knights and liars/switchkins because if this is the case then knights will also have to be present in the third group which contains all three types of people and which says either "we are all liars" or "we are all switchkins". This is not possible as knights never lie.
4) Therefore Knights are present only in the group which says "we are all knights" and this group consists of 10 knights, 10 liars and 10 swithckins. Hence total number of knights is only 10.
5) Now liars never speak the truth and hence they cannot be present in the group which says "we are all liars". Hence liars are present in the group which says "we are all swithchkins" and this group consists of 15 liars and 15 switchkins.
6) The group which says "we are all liars" consists only of switchkins i.e 30 switchkins.
Therefore the three groups are as follows,
1) one in which everybody says "we are all knights" consisting of 10 knights, 10 liars and 10 switchkins.
2) one in which everybody says "we are all switchkins" consisting of 15 liars and 15 switchkins"
3) one in which everybody says "we are all liars" consisting of all switchkins i.e 30 switchkins.
Therefore total number of knights, liars and switchkins are as follows.
knights: 10
Liars: 25
Switchkins:55
solution
