Danny and Robbie draw diagonals of a regular \(2018\)-gon. They can only draw a diagonal that does not cross any other diagonal that has been already drawn, neither it begins nor ends at a same point as any other drawn diagonal. Robbie starts – can he always win?
Adam and Anthony are playing with a chessboard and a rook. The rook can only be moved either to the bottom or to the left. Each of the boys can move it as far as he wants, but only in a straight line either to the bottom or to the left. The boy who places the rook in the bottom left corner wins. Adam starts, show that he can lose to Anthony only if the rook starts somewhere on the main diagonal.
Nathan and Liam have numbers from \(1\) to \(2018\) written on a board. In each move, one of the players removes a number of their choosing, which is still on the board, together with all its remaining divisors. Liam goes first. The last person to remove a number wins. Who has the winning strategy?
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?
Ben and Joe play chess. In addition to a chessboard, they have one rook, which they put in the lower right corner, and they move it in turns. It can only be moved upwards or to the left (for any number of cells). The player who can not make a move, loses. Joe goes first. Who will win with the correct method?