### Brain Teasers

# Truth, Lie or Both ...

Fun: (2.29)
Difficulty: (2.51)
Puzzle ID: #19276

Submitted By: lesternoronha1 Corrected By: brainster

Submitted By: lesternoronha1 Corrected By: brainster

There are three students Anu, Bob and Coy. They have exactly three books - one on Physics, one on Chemistry and one on Math. Each of these three books belongs to exactly one of the students. As I wanted to know the names of the respective owners of the books I asked them about the owners of the books. The following are the answers given by Anu, Bob and Coy.

Anu : Physics book belongs to me. Maths book does not belong to Bob. Chemistry book does not belong to Coy.

Bob : Physics book does not belong to me. Chemistry book belongs to Coy. Maths book does not belong to Anu.

Coy : Physics book does not belong to Anu. Maths book does not belong to me. Chemistry book does not belong to Bob.

I knew that exactly one of them always speaks the truth, exactly one of them always lies and the third one alternates between truth and lie, in any order.

Which of the following statements can then be concluded?

1) Anu always tells the truth.

2) Coy always lies.

3) Bob is the owner of the Chemistry book.

NOTE: Out of the given set of 3 statements, it is not necessary that one and only one statement can be concluded. One or more statements can be concluded.

Anu : Physics book belongs to me. Maths book does not belong to Bob. Chemistry book does not belong to Coy.

Bob : Physics book does not belong to me. Chemistry book belongs to Coy. Maths book does not belong to Anu.

Coy : Physics book does not belong to Anu. Maths book does not belong to me. Chemistry book does not belong to Bob.

I knew that exactly one of them always speaks the truth, exactly one of them always lies and the third one alternates between truth and lie, in any order.

Which of the following statements can then be concluded?

1) Anu always tells the truth.

2) Coy always lies.

3) Bob is the owner of the Chemistry book.

NOTE: Out of the given set of 3 statements, it is not necessary that one and only one statement can be concluded. One or more statements can be concluded.

### Answer

All the three given statements can be concluded.The given statements are

1) Anu always tells the truth.

2) Coy always lies.

3) Bob is the owner of the Chemistry book.

The answers given by Anu, Bob and Coy to my query about the owners of the Physics, Chemistry and Maths books are as follows :-

Anu : Physics book belongs to me. Maths book does not belong to Bob. Chemistry book does not belong to Coy.

Bob : Physics book does not belong to me. Chemistry book belongs to Coy. Maths book does not belong to Anu.

Coy : Physics book does not belong to Anu. Maths book does not belong to me. Chemistry book does not belong to Bob.

Case 1 : Let us assume that Anu always tells the truth.

Then according to Anu's answer,

Physics book belongs to Anu.

Chemistry book belongs to Bob.

Maths book belongs to Coy.

Comparing Anu's reply with that of Bob's and Coy's replies, we can infer that

Bob speaks the truth, lies and speaks the truth respectively when he says that the Physics book does not belong to him, the Chemistry book belongs to Coy and the Maths book does not belong to Anu.

Coy lies three times when he says that the Physics book does not belong to Anu, the Maths book does not belong to him and the Chemistry book does not belong to Bob.

Hence under Case 1, Anu always speaks the truth, Bob alternates between truth and lie and Coy always lies. Also in this case the Chemistry book belongs to Bob.

Case 2 : Let us assume that Bob always tells the truth.

Then according to Bob's answer,

Physics book belongs to Anu.

Maths book belongs to Bob.

Chemistry book belongs to Coy.

Comparing Bob's reply with that of Anu's and Coy's replies, we can infer that

Anu speaks the truth, lies and lies respectively when he says that the Physics book belongs to him, the Maths book does not belong to Bob and the Chemistry book does not belong to Coy.

Coy lies, speaks the truth and speaks the truth respectively when he says that the Physics book does not belong to Anu, the Maths book does not belong to him and the Chemistry book does not belong to Bob.

Hence under Case 2, Bob always speaks the truth, Anu and Coy alternate between truth and lie and hence this case is not possible since it is given that exactly one of them always speaks the truth, exactly one of them always lies and the third one alternates between truth and lie, in any order.

Case 3 : Let us assume that Coy always tells the truth. Here we can again make two sub cases.

a) According to Coy's answer,

Maths book belongs to Anu.

Physics book belongs to Bob.

Chemistry book belongs to Coy.

Comparing Coy's reply with that of Anu's and Bob's replies, we can infer that

Anu lies, speaks the truth and lies respectively when he says that the Physics book belongs to him, the Maths book does not belong to Bob and the Chemistry book does not belong to Coy.

Bob lies, speaks the truth and lies respectively when he says that the Physics book does not belong to him, the Chemistry book belongs to Coy and the Maths book does not belong to Anu.

This case is not possible since both Anu and Bob are alternating between truth and lie and there is no one person who always lies.

b) According to Coy's answer,

Chemistry book belongs to Anu.

Maths book belongs to Bob.

Physics book belongs to Coy.

Comparing Coy's reply with that of Anu's and Bob's replies, we can infer that

Anu lies, lies and speaks the truth respectively when he says that the Physics book belongs to him, the Maths book does not belong to Bob and the Chemistry book does not belong to Coy.

Bob speaks the truth, lies and speaks the truth respectively when he says that the Physics book does not belong to him, the Chemistry book belongs to Coy and the Maths book does not belong to Anu.

This case is also not possible since both Anu and Bob are alternating between truth and lie and there is no one person who always lies.

Hence among the three cases, only Case 1 can be considered to be a valid case wherein Anu always speaks the truth, Bob alternates between truth and lie, Coy always lies and the Chemistry book belongs to Bob. Hence all the given statements 1, 2 and 3 can be concluded.

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