Assume you have a chance to play the following game. You need to put numbers in all cells of a \(10\times10\) table so that the sum of numbers in each column is positive and the sum of numbers in each row is negative. Once you put your numbers you cannot change them. You need to pay £1 if you want to play the game and the prize for completing the task is £100. Is it possible to win?
Once again consider the game from Example 2.
(a) Will you change your answer if the field is a rectangle?
(b) The rules are changed. Now you win if the sum of numbers in each row is greater than 100 and the sum of the numbers in each column is less than 100. Is it possible to win?
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?
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?
Three hedgehogs divided three pieces of cheese of mass of 5g, 8g and 11g. The fox began to help them. It can cut off and eat 1 gram of cheese from any two pieces at the same time. Can the fox leave the hedgehogs equal pieces of cheese?
In the \(4 \times 4\) square, the cells in the left half are painted black, and the rest – in white. In one go, it is allowed to repaint all cells inside any rectangle in the opposite colour. How, in three goes, can one repaint the cells to get the board to look like a chessboard?
On a board there are written 10 units and 10 deuces. During a game, one is allowed to erase any two numbers and, if they are the same, write a deuce, and if they are different then they can write a one. If the last digit left on the board is a unit, then the first player won, if it is a deuce then the second player wins.
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?
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).