Question

(Adapted from [Sm78]) Suppose that on an island there are three types of people, knights, knaves, and normals (also known as spies). Knights always tell the truth, knaves always lie, and normals sometimes lie and sometimes tell the truth. Detectives questioned three inhabitants of the island—Amy, Brenda, and Claire—as part of the investigation of a crime. The detectives knew that one of the three committed the crime, but not which one. They also knew that the criminal was a knight, and that the other two were not. Additionally, the detectives recorded these statements: Amy: “I am innocent.” Brenda: “What Amy says is true.” Claire: “Brenda is not a normal.” After analyzing their information, the detectives positively identified the guilty party. Who was it?

Short answer

Claire is the guilty party.

Step-by-step solution

Introduction

Consider the problem provided in the textbook.

Amy, Brenda, and Claire are three inhabitants of three islands namely knights, knave, and normal.

Assuming Amy to be guilty

If the guilty party is Amy this implies Amy is a knight.

So, Amy always speaks the truth.

But Amy says that I am innocent and this leads to a contradiction.

Assuming Brenda to be guilty

If the guilty party is Brenda this implies Brenda is a knight.

So, Brenda always speaks the truth.

Brenda says that what Amy says is true this implies that what Amy says is true this implies Amy is also a knight but this is a contradiction

Conclusion

Hence, the guilty party is Claire.

This implies Claire is a knight.

So, Claire always speaks the truth.

Since Claire says that Brenda is not normal this implies Brenda is a knave and hence Amy is normal.

So, all the conditions are satisfied in this.

After analyzing their information, the detectives positively identified the guilty party is Claire.