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?
There is a \(5\times 9\) rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?
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?
Karl and Louie are playing the following game. There is a round table that has \(24\) seats around it. Karl and Louie place action figures around the table. However, no two figures are allowed to sit next to each other, regardless if they belong to Karl or Louie. The player, who cannot place their figure loses the game, Karl goes first - show that Louie will always win.
Katie and Andy play the following game: There are \(18\) chocolate bites on a plate. Each player is allowed to take \(1,2\) or \(3\) bites at once. The person who cannot take any more bites loses. Katie starts. Who has the winning strategy?
Arthur and Dan play the following game. There are \(26\) beads on the necklace. Each boy is allowed to take \(1,2,3\) or \(4\) beads at once. The boy who cannot take any more beads loses. Arthur starts - who will win?
Two goblins, Krok and Grok, are playing a game with a pile of gold. Each goblin can take any positive number of coins no larger than \(9\) from the pile. They take moves one after another. There are \(3333\) coins in total, the goblin who takes the last coin wins. Who will win, if Krok goes first?
There are all the numbers from \(1\) to \(2020\) written on the board. Karen and Leon are playing a game where they pick a number off the board and wipe it, together with all of its divisors. Leon goes first - prove that Karen always loses.