Author: I.S. Rubanov
On the table, there are 7 cards with numbers from 0 to 6. Two take turns in taking one card. The winner is the one is the first person who can, from his cards, make up a natural number that is divisible by 17. Who will win in a regular game the person who goes first or second?
Hannah Montana wants to leave the round room which has six doors, five of which are locked. In one attempt she can check any three doors, and if one of them is not locked, then she will go through it. After each attempt her friend Michelle locks the door, which was opened, and unlocks one of the neighbouring doors. Hannah does not know which one exactly. How should she act in order to leave the room?
There are 30 students in a class: excellent students, mediocre students and slackers. Excellent students answer all questions correctly, slackers are always wrong, and the mediocre students answer questions alternating one by one correctly and incorrectly. All the students were asked three questions: “Are you an excellent pupil?”, “Are you a mediocre student?”, “Are you a slacker?”. 19 students answered “Yes” to the first question, to the second 12 students answered yes, to the third 9 students answered yes. How many mediocre students study in this class?
In the numbers of MEXAILO and LOMONOSOV, each letter denotes a number (different letters correspond to different numbers). It is known that the products of the numbers of these two words are equal. Can both numbers be odd?
On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?
Solving the problem: “What is the solution of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^3 + 2000x^2 + 2000x + 3000\) (\(x\) is a real number) if \(x^2 + x + 1 = 0\)?”, Vasya got the answer of 3000. Is Vasya right?
In the dense dark forest ten sources of dead water are erupting from the ground: named from #1 to #10. Of the first nine sources, dead water can be taken by everyone, but the source #10 is in the cave of the dark wizard, from which no one, except for the dark wizard himself, can collect water. The taste and color of dead water is no different from ordinary water, however, if a person drinks from one of the sources, then he will die. Only one thing can save him: if he then drinks poison from a source whose number is greater. For example, if he drinks from the seventh source, then he must necessarily drink poison from the #8, #9 or #10 sources. If he doesn’t drink poison from the seventh source, but does from the ninth, only the poison from the source #10 will save him. And if he originally drinks the tenth poison, then nothing will help him now. Robin Hood summoned the dark wizard to a duel. The terms of the duel were as follows: each brings with him a mug of liquid and gives it to his opponent. The dark wizard was delighted: “Hurray, I will give him poison #10, and Robin Hood can not be saved! And I’ll drink the poison, which Robin Hood brings to me, then ill drink the #10 poison and that will save me!” On the appointed day, both opponents met at the agreed place. They honestly exchanged mugs and drank what was in them. However, afterwards erupted the joy and surprise of the inhabitants of the dark forest, when it turned out that the dark wizard had died, and Robin Hood remained alive! Only the Wise Owl was able to guess how Robin Hood had managed to defeat dark wizard. Try and guess as well.
There are five chain links with 3 rings in each. What is the smallest number of rings that need to be unhooked and hooked together to connect these links into one chain?
In Wonderland, an investigation was conducted into the case of a stolen soup. At the trial, the White Rabbit said that the soup was stolen by the Mad Hatter. The Cheshire Cat and the Mad Hatter also testified, but what they said, no one remembered, and the record was washed away by Alice’s tears. During the court session, it became clear that only one of the defendants had stolen the soup and that only he had given a truthful testimony. So, who stole the soup?
The stepmother, leaving for the ball, gave Cinderella a sack which contained a mixture of poppy and millet, and ordered them to be sorted. When Cinderella was leaving for the ball, she left three sacks: one contained millet, the other contained poppy, and in the third – a mixture that had not yet been sorted. In order not to confuse the sacks, Cinderella attached a label to each of them that said: “Poppy seed”, “Millet” and “Mixture”. The stepmother returned from the ball first and deliberately swapped all of the labels in such a way that on each sack there was an incorrect inscription. The fairy godmother managed to warn Cinderella that now none of the labels on the sacks were correct. Then Cinderella took out only one single grain from one sack and, looking at it, immediately guessed what was in each sack. How did she do this?