A journalist came to a company which had \(N\) people. He knows that this company has a person \(Z\), who knows all the other members of the company, but nobody knows him. A journalist can address each member of the company with the question: “Do you know such and such?” Find the smallest number of questions sufficient to surely find \(Z\). (Everyone answers the questions truthfully. One person can be asked more than one question.)
Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes (if the time to turn over and transfer the cutlets is neglected)?
Your task is to find out a five-digit phone number, asking questions that can be answered with either “yes” or “no.” What is the smallest number of questions for which this can be guaranteed (provided that the questions are answered correctly)?
What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.
The rook stands on the square a1 of a chessboard. For a move, you can move it by any number of cells to the right or up. The one who puts the rook on the h8 square will win. Who wins with the right strategy?
Hannah has 10 employees. Each month, Hannah raises the salary by 1 pound of exactly nine of her employees (of her choice). How can Hannah raise the salaries to make them equal? (Salaries are an integer number of pounds.)
Three friends decide, by a coin toss, who goes to get the juice. They have one coin. How do they arrange coin tosses so that all of them have equal chances to not have to go and get the juice?
To transmit messages by telegraph, each letter of the Russian alphabet () ( and are counted as identical) is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet (letter numbering starts from zero). For example, the letter is represented in the form 00000, letter -00001, letter -10111, letter -11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters is received. Find the word you sent.
In one urn there are two white balls, in another two black ones, in the third – one white and one black. On each urn there was a sign indicating its contents: WW, BB, WB. Someone rehung the signs so that now each sign indicating the contents of the urn is incorrect. It is possible to remove a ball from any urn without looking into it. What is the minimum number of removals required to determine the composition of all three urns?
a) There are 21 coins on a table with the tails side facing upwards. In one operation, you are allowed to turn over any 20 coins. Is it possible to achieve the arrangement were all coins are facing with the heads side upwards in a few operations?
b) The same question, if there are 20 coins, but you are allowed to turn over 19.