1. There is a group of 3 people, X, Y, and Z. Each one may be either an honest person or a deceiver. An honest person tells the truth. A deceiver always tells the falsehood. Suppose that you ask X, "Are you a deceiver?" X responds to you but you do not hear his response. Y says, "X denies that he is a deceiver." Z says, "X is a deceiver." How many deceivers are there in the group?

2. Students ask their history teacher what topics will appear in the final exam. The teacher tells the students the following: –1) Either World War I or World War II, but not both. –2) Either World War II and French revolution or none of them. –3) If World War I does not appear in the exam, French revolution will not either. Which topic will appear in the exam?

3. A, B, C, D are 3 bank-tellers who are taking a test in distinguishing fake banknotes. They are asked to exam 3 banknotes to determine whether they are genuine or fake. –A: The 1st one is genuine but the 3rd one is fake. –B: The 2nd and the 3rd are fake. –C: The 1st is genuine, the 2nd and 3rd cannot both be genuine or both be fake. The examiner says that each of them is half right. That means their answer is no better than by chance. Without examining the banknotes, D can tell which one is genuine and which is fake. Explain.

4. There are 5 hats: 3 of them are black and 2 of them are white. 3 of these hats were worn by 3 people who are standing in line facing the wall. Each of them can only see the hat(s) of the one(s) standing before him (except the first person, who is facing the wall ). We ask the one who standing in the last position whether he knows the color of his own hat. His reply is negative. Then we ask the one who is standing in the middle position the same question. His reply is also negative. However, the first person knows the color of his hat. How does he know?