Problems

On a table, there are five coins lying in a row: the middle one lies with a head facing upwards, and the rest lie with the tails side up. It is allowed to simultaneously flip three adjacent coins. Is it possible to make all five coins positioned with the heads side facing upwards with the help of several such overturns?

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?

In a bookcase, there are four volumes of the collected works of Astrid Lindgren, with each volume containing 200 pages. A worm who lives on this bookshelf has gnawed its way from the first page of the first volume to the last page of the fourth volume. Through how many pages has the worm gnawed its way through?

There are some incorrect weighing scales, a bag of cereal and a correct weight of 1 kg. How can you weigh on these scales 1 kg of cereals?

Michael thinks of a number no less than \(1\) and no greater than \(1000\). Victoria is only allowed to ask questions to which Michael can answer “yes” or “no” (Michael always tells the truth). Can Victoria figure out which number Michael thought of by asking \(10\) questions?

Can the equality \(K \times O \times T = U \times W \times E \times N \times H \times Y\) be true if the numbers from 1 to 9 are substituted for letters instead of the letters? Different letters correspond to different numbers.

In a room, there are 85 balloons – red and blue ones. It is known that: 1) at least one of the balls is red, 2) from each arbitrarily chosen pair of balls at least one is blue. How many red balls are there in the room?

There are 6 locked suitcases and 6 keys to them. At the same time, it is not known to which suitcase each key fits. What is the smallest number of attempts you need to make in order to open all the suitcases for sure? And how many attempts will it take there are not 6 but 10 keys and suitcases?

If the Humpbacked Horse does not eat or sleep for seven days, it will lose its magical powers. Suppose he did not eat or sleep for a week. What should he do first of all by the end of the seventh day – eat or sleep, so as not to lose powers?

Decipher the following rebus (see the figure). Despite the fact that only two figures are known here, and all others are replaced by asterisks, the example can be restored.