Problems

A rectangular chocolate bar size $5\times 10$ is divided by vertical and horizontal division lines into 50 square pieces. Two players are playing the following game. The one who starts breaks the chocolate bar along some division line into two rectangular pieces and puts the resulting pieces on the table. Then players take turns doing the same operation: each time the player whose turn it is at the moment breaks one of the parts into two parts. The one who is the first to break off a square slice $1\times 1$ (without division lines) a) loses; b) wins. Which of the players can secure a win: the one who starts or the other one?

Two players in turn paint the sides of an $n$-gon. The first one can paint the side that borders either zero or two colored sides, the second – the side that borders one painted side. The player who can not make a move loses. At what $n$ can the second player win, no matter how the first player plays?

The March Hare and the Dormouse also decided to play a game. They made two piles of matches on the table. The first pile contains 7 matches, and the second one 8. The March Hare set the rules: the players divide a pile into two piles in turns, i.e. the first player divides one of the piles into two, then the second player divides one of the piles on the table into two, then the first player divides one of the piles into two and so on. The loser is the one who cannot not find a pile to divide. The March Hare starts the game. Can the March Hare play in such a way that he always wins?

A board $7\times 7$ is coloured in chessboard fashion in such a way that all the corners are black. The Queen orders the Hatter to colour the board white but sets the rule: in one go it is allowed to repaint only two adjacent cells into opposite colours. The Hatter tries to explain that this is impossible. Can you help the Hatter to present his arguments?

The Hatter plays a computer game. There is a number on the screen, which every minute increases by 102. The initial number is 123. The Hatter can change the order of the digits of the number on the screen at any moment. His aim is to keep the number of the digits on the screen below four. Can he do it?

The Hatter has a peculiar ancient device, which can perform the following three operations: for each $x$ and $y$ it calculates $x+y$, $x-y$ and $\frac{1}{x}$ (for $x\ne 0$).

(a) The Hatter claims that he can square any positive real number using the device by performing not more than 6 operations. How can he do it?

(b) Moreover, the Hatter claims that he can multiply any two positive real numbers with the help of the device by performing not more than 20 operations. Can you show how?

(All intermediate results are allowed to be written down, and can be used in further calculations.)

Show that in the game “Noughts and Crosses” the second player never wins if the first player is smart enough.

There is a chequered board of dimension $10\times 12$. In one go you are allowed to cross out any row or column if it contains at least one square which was not crossed out yet. The loser is the player who cannot make a move. Is there a winning strategy for any player?

Pathways in the Wonderland zoo make a equilateral triangles with middle lines drawn. A monkey has escaped from it’s cage. Two zoo caretakers are catching the monkey. Can zookeepers catch the monkey if all three of them are running only on pathways, the running speeds of the monkey and the zookeepers are equal, and they are all able to see each other?

There is a chequered board of dimension (a) $9\times 10$, (b) $9\times 11$. In one go you are allowed to cross out any row or column if it contains at least one square which was not crossed out yet. The loser is the player who cannot make a move. Is there a winning strategy for any player?