Problems

The numbers 25 and 36 are written on a blackboard. Consider the game with two players where: in one turn, a player is allowed to write another natural number on the board. This number must be the difference between any two of the numbers already written, such that this number does not already appear on the blackboard. The loser is the player who cannot make a move.

Consider a chessboard of size (number of rows $\times$ number of columns): a) $9\times 10$; b) $10\times 12$; c) $9\times 11$. Two people are playing a game where: in one turn a player is allowed to cross out any row or column as long as there it contains at least one square that is not crossed out. The loser is the player who cannot make a move. Which player will win?

Two players in turn put coins on a round table, in such a way that they do not overlap. The player who can not make a move loses.

Two people take turns placing bishops on a chessboard such that the bishops cannot attack each other. Here, the colour of the bishops does not matter. (Note: bishops move and attack diagonally.) Which player wins the game, if the right strategy is used?

There are two piles of rocks, each with 7 rocks. Consider the game with two players where: in one turn you can take any amount of rocks, but only from one pile. The loser is the one who has no rocks left to take.

Two people take turns placing knights on a chessboard such that the knights cannot attack each other. The loser is the player who cannot make a move. Which player wins the game, if the right strategy is used?

Two people take turns placing kings on squares of a $9\times 9$ chessboard such that the kings cannot attack each other. The loser is the player who cannot make a move. Which player wins the game, if the right strategy is used?

a) Two in turn put bishops in the cells of a chessboard. The next move must beat at least one empty cell. The bishop also beats the cell in which it is located. The player who loses is the one who cannot make a move.

b) Repeat the same, but with rooks.

There is a board of $10\times 10$ squares. Consider the game with two players where: in one turn a player is allowed to cover any two adjacent squares with a domino (a $1\times 2$ rectangle) so that the domino doesnt cover another domino. The loser is the one who cannot make a move. Which player would win, if the right strategy was used?

In each square of an $11\times 11$ board there is a checker. Consider the game with two players where: in one move a player is allowed to take any amount of adjacent checkers from the board, as long as they checkers are in the same vertical column or in the same horizontal row. The winner is the player who removes the last checker. Which player wins the game?