Alex and Priyanka have a chessboard and a queen on it. Each of the players can only move the queen to the top, to the right, or along a diagonal – to the top and right (like the queen moves, but only in three directions out of all eight). The person who places the queen in the top right corner wins. The chessboard is a normal \(8 \times 8\) board. The queen starts four squares to the right from the bottom left corner. If Priyanka starts, who will win the game?
Can you cover a \(10 \times 10\) board using only \(T\)-shaped tetraminoes?
Can you cover a \(10 \times 10\) square with \(1 \times 4\) rectangles?
Two opposite corners were removed from an \(8 \times 8\) chessboard. Can you cover this chessboard with \(1 \times 2\) rectangular blocks?
One small square of a \(10 \times 10\) square was removed. Can you cover the rest of it with 3-square \(L\)-shaped blocks?
A \(7 \times 7\) square was tiled using \(1 \times 3\) rectangular blocks. One of the squares has not been covered. Which one can it be?
Can you cover a \(13 \times 13\) square using two types of blocks: \(2 \times 2\) squares and \(3 \times 3\) squares?
Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.
A two-player game with matches. There are 37 matches on the table. In each turn, a player is allowed to take no more than 5 matches. The winner of the game is the player who takes the final match. Which player wins, if the right strategy is used?
The seller with weights. With four weights the seller can weigh any integer number of kilograms, from 1 to 40 inclusive. The total mass of the weights is 40 kg. What are the weights available to the seller?