On a board of size \(8 \times 8\), two in turn colour the cells so that there are no corners of three coloured squares. The player who can’t make a move loses. Who wins with the right strategy?
On a plane there are 100 sheep-points and one wolf-point. In one move, the wolf moves by no more than 1, after which one of the sheep moves by a distance of no more than 1, after that the wolf again moves, etc. At any initial location of the points, will a wolf be able to catch one of the sheep?
A \(99 \times 99\) chequered table is given, each cell of which is painted black or white. It is allowed (at the same time) to repaint all of the cells of a certain column or row in the colour of the majority of cells in that row or column. Is it always possible to have that all of the cells in the table are painted in the same colour?
10 guests came to a party and each left a pair of shoes in the corridor (all guests have the same shoes). All pairs of shoes are of different sizes. The guests began to disperse one by one, putting on any pair of shoes that they could fit into (that is, each guest could wear a pair of shoes no smaller than his own). At some point, it was discovered that none of the remaining guests could find a pair of shoes so that they could leave. What was the maximum number of remaining guests?
How can one measure out 15 minutes, using an hourglass of 7 minutes and 11 minutes?
Two boys play the following game: they take turns placing rooks on a chessboard. The one who wins is the one whose last move leaves all the board cells filled. Who wins if both try to play with the best possible strategy?
In a basket there are 13 apples. There are scales, with which you can find out the total weight of any two apples. Think of a way to find out from 8 weighings the total weight of all the apples.
A daisy has a) 12 petals; b) 11 petals. Consider the game with two players where: in one turn a player is allowed to remove either exactly one petal or two petals which are next to each other. The loser is the one who cannot make a turn. How should the second player act, in cases a) and b), in order to win the game regardless of the moves of the first player?
On the board the number 1 is written. Two players in turn add any number from 1 to 5 to the number on the board and write down the total instead. The player who first makes the number thirty on the board wins. Specify a winning strategy for the second player.
There are two stacks of coins on a table: in one of them there are 30 coins, and in the other – 20. You can take any number of coins from one stack per move. The player who cannot make a move is the one that loses. Which player wins with the correct strategy?