Prove that in any group of 10 whole numbers there will be a few whose sum is divisible by 10.
You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
a) A piece of wire that is 120 cm long is given. Is it possible, without breaking the wire, to make a cube frame with sides of 10 cm?
b) What is the smallest number of times it will be necessary to break the wire in order to still produce the required frame?
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 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?