Two people take turns drawing noughts and crosses on a \(9 \times 9\) grid. The first player uses crosses and the second player uses noughts. After they finish, the number of rows and columns where there are more crosses than noughts are counted, and these are the points which the first player receives. The number of rows and columns where there are more noughts than crosses are the second player’s points. The player who has the most points is the winner. Who wins, if the right strategy is used?
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?
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.
It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
An adventurer is travelling to the planet of liars and truth tellers with an official guide and is introduced to a local. “Are you a truth teller?” asked the adventurer. The alien answers “Yrrg,” which means either “yes” or “no”. The adventurer asks the guide for a translation. The guide says “"yrrg" means "yes". I will add that the local is actully a liar.” Is the local alien liar or truth teller?
In a certain realm there are magicians, sorcerers and wizards. The following is known about them: firstly, not all magicians are sorcerers, and secondly, if the wizard is not a sorcerer, then he is not a magician. Is it true that not all magicians are wizards?
A traveller on the planet of liars and truth tellers met four people and asked them: “Who are you?”. They received the following answers:
1st: “We are all liars.”
2nd: “Among us is exactly one liar.”
3rd: “Among us there are two liars.”
4th: “I have never lied and I’m not lying”.
The traveller quickly realised who the fourth resident was. How did they do it?
In the lower left corner of an 8 by 8 chessboard is a chip. Two in turn move it one cell up, right or right-up diagonally. The one who puts the chip in the upper right corner wins. Who will win in a regular game?