1. Students in Class
A problem has been proposed in class. At the end of the lesson it turned out that the number of boys, who had solved the problem, was the same as the number of girls, who had not solved it. Were there more girls in the class than students who had solved the problem?
Answer: The number of all the girls in the class equals to the number of students who solved the problem. Indeed, by the condition, the number of students who solved the problem equals to the sum of the number of girls who solved it and the number of boys, who solved it. But the number of boys who solved the problem is the same as the number of girls who did not solve the problem. So, the number of all the girls in the class equals to the number of students who solved the problem.
2. Are you a liar?
Robinson found himself on an island where some of the people were liars, and others always told the truth. When he met with one of the inhabitant of the island, he asked him: “Are you a liar or not?” “I’m not a liar”, answered the person. “All right, if it is so, you’ll be my companion”, Robinson said. After a while they saw another man. Robinson pointed to the man and asked his new friend, “Could you, please, ask him, if he is a liar or not?” The new friend asked the question to the man, came back and said, “He said he was not a liar”. “All right, now I’m convinced that you are not a liar!” smiled Robinson. What convinced Robinson?
Answer: Regardless of who is interrogated (liar or not), to the question “Are you a liar or not?” the person will answer: “I’m not a liar!”. Indeed, if he is not a liar, then he is telling the truth. If he is a liar, then he will say a lie about himself (that is, that he is not a liar). So, Robinson was just checking the honesty of his companion: if the companion was a liar, then he would lie and tell Robinson that the answer was “I’m a liar!”. Since the companion said that the answer was “I’m not a liar!”, then the companion must
be a truth-teller.
3. Defective Box
10 cigarette boxes each having 10 cigarettes are placed on a table. Each cigarette weighs 10 gram in a normal box but one defective box has each cigarette weighing 9 grams in it. You are allowed to use a “weighing machine” only once and tell which box is defective.
Answer: Take 1 cigarette from 1st box, 2 cigarettes from 2nd box, 3 cigarettes from 3rd box and so on till 10 cigarettes from 10thbox. Now weigh them. The (550-weight of total cigarettes)th box is defective.