Consider a rectangular parallelepiped with size a) \(4 \times 4 \times 4\); b) \(4 \times 4 \times 3\); c) \(4 \times 3 \times 3\), made up of unit cubes. Consider the game with two players where: in one turn a player is allowed to pierce through any row with a long wire, as long as there is at least one cube in the row with no wire. The loser is the player who cannot make a move. Who would win, if the right strategy is used?
Two people take turns drawing noughts and crosses on a \(9 \times 9\) grid. The first player uses crosses and the second player uses noughts. After they finish, the number of rows and columns where there are more crosses than noughts are counted, and these are the points which the first player receives. The number of rows and columns where there are more noughts than crosses are the second player’s points. The player who has the most points is the winner. Who wins, if the right strategy is used?
A rook is on the a1 square of a chessboard. Consider the game with two players where: in one move a player can move the rook by any number of squares to the left, right or up. The winner is the player who places the rook on the square h8. Who would win, if the right strategy is used?
There are two piles of sweets: one with 20 sweets and the other with 21 sweets. In one go, one of the piles needs to be eaten, and the second pile is divided into two not necessarily equal piles. The player that cannot make a move loses. Which player wins and which one loses?
The game begins with the number 0. In one go, it is allowed to add to the actual number any natural number from 1 to 9. The winner is the one who gets the number 100.
Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.
An \(8 \times 8\) square is painted in two colours. You can repaint any \(1 \times 3\) rectangle in its predominant colour. Prove that such operations can make the whole square monochrome.
Some person \(A\) thought of a number from 1 to 15. Some person \(B\) asks some questions to which you can answer ‘yes’ or ‘no’. Can \(B\) guess the number by asking a) 4 questions; b) 3 questions.
The numbers from 1 to 9999 are written out in a row. How can I remove 100 digits from this row so that the remaining number is a) maximal b) minimal?
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?