Question

A says “We are both knaves” and B says nothing. Exercises 24–31 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 (called normals by Sullying [Sm78]) 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 says “I am the knight,” B says, “A is not the knave,” and C says “B is not the knave.”

Short answer

A is a knave B is a spy and C is a knight.

Step-by-step solution

Tips

There are inhabitants of an island on which there are three kinds of people:

• Knights who always tell the truth
• Knaves who always lie
• Spies who can either lie or tell the truth.

The truth value for given Statement

Given: One knight, one knave and one spy.

Knight: always tells the truth

Knave: always lies

Spy: lies or tells the truth

A = "I am the knight".

B = "A is not the knave".

C = "B is not the knave".

Let us first assume that A is a knight, then A is telling the truth and B is also telling the truth. Since B is telling the truth and B cannot be a knight (since A is already the knight), then B must be the spy. The only remaining position for C is then the knave, but then B has to be a knave (while we know he's a spy) and thus we obtained a contradiction. Thus the assumption is invalid and thus A is not a knight.

Let us next assume that B is a knight, then B is telling the truth andC is telling the truth. Since C is telling the truth and C cannot be a knight (since B is already the knight), then C must be the spy. However, A cannot be a knave (by B 's true statement) while the only position left for A is a knave, thus we have obtained a contradiction. Thus the assumption is invalid and thus B is not a knight.

Since A and B are no knights, C then has to be the knight. C Tells the truth, thus B is not the knave and thus A has to be the knave, while B is the spy (only remaining position).