Problems

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 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.)

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?

The March Hare and the Dormouse are playing a game. A rook is placed on square a1 on a chessboard. In one go it is allowed to move the rook by any number of squares but only up or to the right. The winner is the one who places the rook on square h8. The Dormouse makes the first move. Who will win the game? (It is assumed that everybody is following the best possible strategy).

The March Hare made three piles of stones of 10, 15, and 20 stones respectively, and invited the Dormouse to play the following game. It is allowed to split any existing pile into two smaller ones in one go. The loser is the one who cannot make a move.

Alice and the Hatter decided to play another game. They found a field with exactly 2016 stones on it. In one go Alice picks 1 or 4 stones, while the Hatter picks 1 or 3 stones. The loser is the one who cannot make a move. Can Alice or the Hatter win irrespective of the other player’s strategy?

Tweedledum and Tweedledee play a game. They have written numbers 1, 2, 3, 4 in a circle. Tweedledum, who makes the first move, can add 1 to any two adjacent numbers; while Tweedledee is allowed to exchange any two adjacent numbers. Tweedledum wins if all the numbers become equal. Can Twedleedee prevent Tweedledum from winning if both must make a move every turn?

Assume you have a chance to play the following game. You need to put numbers in all cells of a $10\times 10$ table so that the sum of numbers in each column is positive and the sum of numbers in each row is negative. Once you put your numbers you cannot change them. You need to pay £1 if you want to play the game and the prize for completing the task is £100. Is it possible to win?

Once again consider the game from Example 2.

(a) Will you change your answer if the field is a rectangle?

(b) The rules are changed. Now you win if the sum of numbers in each row is greater than 100 and the sum of the numbers in each column is less than 100. Is it possible to win?