This time, Sally and Fatima have some number of books on a shelf. Every turn, each of them is allowed to take 1, 3 or 4 books from the shelf. The girl that takes the last book wins, Sally goes first. Who will win if there are: a) 14, b) 16, c) 19 books on the shelf?
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?
A chequered strip of \(1 \times N\) is given. Two players play the game. The first player puts a cross into one of the free cells on his turn, and subsequently the second player puts a nought in another one of the cells. It is not allowed for there to be two crosses or two noughts in two neighbouring cells. The player who is unable to make a move loses.
Which of the players can always win (no matter how their opponent played)?
Hannah and Emma have three coins. On different sides of one coin there are scissors and paper, on the sides of another coin – a rock and scissors, on the sides of the third – paper and a rock. Scissors defeat paper, paper defeats rock and rock wins against scissors. First, Hannah chooses a coin, then Emma, then they throw their coins and see who wins (if the same image appears on both, then it’s a draw). They do this many times. Is it possible for Emma to choose a coin so that the probability of her winning is higher than that of Hannah?
A White Rook pursues a black bishop on a board of \(3 \times 1969\) cells (they walk in turn according to the usual rules). How should the rook play to take the bishop? White makes the first move.
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?
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?