Problems

A two-player game with matches. There are 37 matches on the table. In each turn, a player is allowed to take no more than 5 matches. The winner of the game is the player who takes the final match. Which player wins, if the right strategy is used?

Multiplication of numbers. Restore the following example of the multiplication of natural numbers if it is known that the sum of the digits of both factors is the same.

Restore the example of the multiplication.

Petya and Misha play such a game. Petya takes in each hand a coin: one – 10 pence, and the other – 15. After that, the contents of the left hand are multiplied by 4, 10, 12 or 26, and the contents of the right hand – by 7, 13, 21 or 35. Then Petya adds the two results and tells Misha the result. Can Misha, knowing this result, determine which hand – the right or left – contains the 10 pence coin?

There are some weighing scales without weights and 3 identical in appearance coins, one of which is fake: it is lighter than a real coin (real coins are equal in weight). How many weighings are needed to determine a counterfeit coin?

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?

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?

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?

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.