**DEMOLESSON**

**Logical problems involving false statements. **

There are logic problems where some statements are false or incorrect, and your job is to figure out what is true and what is false. To solve such problems you should make assumptions.

*Example:* Suppose we have 3 statements, one of which is false. *Statement 1: *"A is white"; *Statement 2: *"A is not white" *Statement 3:* "A is grey".

Can you say what color A is?

To solve a problem, make a table. Assume that A is white (first statement) is true. Then statements 2 and 3 are false, but there is only one false statement. So assumption is wrong. Assume that A is not white. Statement 1 is false, but we cannot tell about statement 3. So second assumption does not work either. Assume that A is grey. Then statement 2 is true and statement 1 is false.

Statement 1 |
if T |
F |
F |

Statement 2 |
F |
if T |
T |

Statement 3 |
F |
may be T or F |
if T |

**Exercises.**

1. Three friends - Tom, Rick and Dan - played soccer in the yard, and one boy accidentally broke the window in the house. When Dan's mom came, he said, "Rick broke the window". Tom said, "I did not break the window". Only one of the boys told the truth. Who broke the window?

2. In the forest, animal athletes ran a 100 m race. After the race was over, spectators were discussing the results. One squirrel said, "Hare got the first place, and Fox got the second place". Second squirrel said, "Moose was first, Hare was second". After that Owl remarked that only one part of each squirrel's statement was true. Can you tell from it who got the first place in the race and who got the second place?

3. Three boys, Petya, Vasya and Kolya discussed a present for their friend Tolya. Petya said, "Let us buy Tolya a book. He loves books. He has at least 100 books." Vasya said, "I think he has less than 100 books". Kolya said, "I do not know how many books Tolya has, but he has at least one book." When boys went to Tolya's birthday party, it turned out that only one of them was right about number of books that Tolya had. How many books did Tolya have before his birthday?

4. There are two white balls in one box, one black ball and one white ball in the second box and two black balls in the third box. All the drawings on the boxes show what balls are inside incorrectly. From what box can you take out just one ball without looking and then tell what balls are inside each box? Explain your answer.

**5. Country of Knights and Liars. **

The problems in this section are about so called Country of Knights and Liars. Two kinds of people live in this country: knights and liars. Knights always tell the truth; liars always lie. Also you can meet tourists there who are not citizens of this country. They can either lie or tell the truth. A smart observer can always figure out who is a knight, who is a liar and who is a tourist.

a. One day I met a person in the Country of Knights and Liars and he said about himself, "I am a liar". Who was he?

b. Here are three citizens of the Country of Knights and Liars - Am, Bom and Kim. Am says, "We are all liars". Bom says, "Only one of us is a liar". Who is Kim, liar of knight? Is it possible to determine who Bom is?

c. One day I asked everybody in a group of 6 citizens of this country who knew each other: "How many of you are knights?" I received the following answers: 0, 1, 2, 3, 4, 5. How many of them were knights?

d. There are 12 ministers in the government of the Country of Knights and Liars. They are all citizens of the country. Some of them are liars, some of them are knights.

Once, at a meeting, one of the ministers exclaims, "There are no knights among us!"

Second minister says, "There is **no more than** 1 knight here";

Third minister says, "There are **no more than **2 knights here";

Fourth minister says, "There are** no more than** 3 knights here";

and so on, until twelfth minister says, "There are **no more than** 11 knights here".

How many knights and how many liars are there in the government?

e. Once I overheard a conversation among four citizens of the Country of Knights and Liars.

Tim said, **"At least** one of us is a liar";

Sam said, **"At least** two of us are liars";

Pam said, **"At least** three of us are liars";

Jim said, "There are no liars among us".

But there were liars among them. How many and who?

f. Two criminals robbed a train. Police arrested six suspects T, U,V, X,Y and Z. The two robbers are among the suspects. When asked about the robbery each suspect makes a statement. Two guilty ones lie, others tell the truth. Who robbed the train?

T : V did it.

U : Z did not do it .

V : X did not do it.

X : U did it.

Y : T did not do it.

Z : Y did not do it.