Answer to Question #131740 in Discrete Mathematics for assignment

Question

The following exercises relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth,knaves who always lie, and spies who can either lie or tell the truth. You encounter three people,A,B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions.
(a)A says “I am the knave”,B says “I am the knave”, and C says “I am the knave”.
b)A says “I am the knight”,B says “A is not the knave”, and C says“B is not the knave”.
(c)A says “I am not the spy”,B says “I am not the spy”, and C says“A is the spy”.

Answer 1

(a). Knights always tell the truth, so they cannot say "I'm a knave". Knaves always lie. If knave say "I'm a knave" it will be truth, so knaves have say only "I'm a knight". Since all 3 persons cannot be neither knights, nor knaves, they are spies.

(b). “I am the knight” can belong to knight (truth), knave(lie), or spy (lie). “A is not the knave” can belong to knight if A is knight or spy, or it can belong to knave (with any A), or spy (with any A). In this case there is no unique solution. Possible solutions for A,B,C are: {(knight, knight, knight), (knight, knight, spy), (knight, spy, knight), (knight, spy, spy), (knave, spy, knight), (knave, spy, spy), (knave, knave, knave)}.

(c). If knave says "I'm a spy", then he says truth, so knaves cannot say this. “A is the spy” can be said by knight, by spy (if A is a knight or a spy) or by knave (if A is a knight). All possible solutions for A, B, C are: {(knight, knight, spy), (knight, spy, spy), (spy, knight, knight), (spy, spy,spy), (spy, spy, knight), (knight, knight, knave), (knight, spy, knave)}.