Two players take turns to put rooks on a chessboard so that the rooks cannot capture each other. The player who cannot make a move loses.
On a board there are written 10 units and 10 deuces. During a game, one is allowed to erase any two numbers and, if they are the same, write a deuce, and if they are different then they can write a one. If the last digit left on the board is a unit, then the first player won, if it is a deuce then the second player wins.
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.
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.
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?
There are two piles of rocks: one with 30 rocks and the other with 20 rocks. In one turn a player is allowed to take any number of rocks but only from one of the piles. The loser is the player who has no rocks left to take. Who would win in a two player game, if the right strategy is used?
There are twenty dots distributed along the circumference of circle. Consider the game with two players where: in one move a player is allowed to connect any two of the dots with a chord (aline going through the inside of the circle), as long as the chord does not intersect those previously drawn. The loser is the one who cannot make a move. Which player wins?
There are two piles of sweets: one with 20 sweets and the other with 21 sweets. In one go, one of the piles needs to be eaten, and the second pile is divided into two not necessarily equal piles. The player that cannot make a move loses. Which player wins and which one loses?
The game begins with the number 0. In one go, it is allowed to add to the actual number any natural number from 1 to 9. The winner is the one who gets the number 100.
Prove that: \[a_1 a_2 a_3 \cdots a_{n-1}a_n \times 10^3 \equiv a_{n-1} a_n \times 10^3 \pmod4,\] where \(n\) is a natural number and \(a_i\) for \(i=1,2,\ldots, n\) are the digits of some number.