Two grandmasters in turn put rooks on a chessboard (one turn – one rook) so that they cannot capture each other. The person who cannot put a rook on the chessboard loses. Who will win with the game – the first or second grandmaster?
Given a board (divided into squares) of the size: a) \(10\times 12\), b) \(9\times 10\), c) \(9\times 11\), consider the game with two players where: in one turn a player is allowed to cross out any row or any column if there is at least one square not crossed out. The loser is the one who cannot make a move. Is there a winning strategy for one of the players?
One and a half diggers dig for a half hour and end up having dug half a pit. How many pits will two diggers dig in two hours?
A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?
Let \(x\) be a natural number. Among the statements:
\(2x\) is more than 70;
\(x\) is less than 100;
\(3x\) is greater than 25;
\(x\) is not less than 10;
\(x\) is greater than 5;
three are true and two are false. What is \(x\)?
A broken calculator carries out only one operation “asterisk”: \(a*b = 1 - a/b\). Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).
A message is encrypted by replacing the letters of the source text with pairs of digits according to some table (known only to the sender and receiver) in which different letters of the alphabet correspond to different pairs of digits. The cryptographer was given the task to restore the encrypted text. In which case will it be easier for him to perform the task: if it is known that the first word of the second line is a “thermometer” or that the first word of the third line is “smother”? Justify your answer. (It is assumed that the cryptographic table is not known).
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.
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.
Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?