There is a chocolate bar with five longitudinal and eight transverse grooves, along which it can be broken (in total into \(9 * 6 = 54\) squares). Two players take part, in turns. A player in his turn breaks off the chocolate bar a strip of width 1 and eats it. Another player who plays in his turn does the same with the part that is left, etc. The one who breaks a strip of width 2 into two strips of width 1 eats one of them, and the other is eaten by his partner. Prove that the first player can act in such a way that he will get at least 6 more chocolate squares than the second player.
Two play tic-tac-toe on a \(10 \times 10\) board according to the following rules. First they fill the whole board with noughts and crosses, putting them in turn (the first player puts crosses, their partner – noughts). Then two numbers are counted: \(K\) is the number of five consecutively standing crosses and \(H\) is the number of five consecutively standing zeros. (Five, standing horizontally, vertically and parallel to the diagonal are counted, if there are six crosses in a row, this gives two fives, if there are seven, then three, etc.). The number \(K-H\) is considered to be the winnings of the first player (the losses of the second).
a) Does the first player have a winning strategy?
b) Does the first player have a non-losing strategy?
A game takes place on a squared \(9 \times 9\) piece of checkered paper. Two players play in turns. The first player puts crosses in empty cells, its partner puts noughts. When all the cells are filled, the number of rows and columns in which there are more crosses than zeros is counted, and is denoted by the number \(K\), and the number of rows and columns in which there are more zeros than crosses is denoted by the number \(H\) (18 rows in total). The difference \(B = K - H\) is considered the winnings of the player who goes first. Find a value of B such that
1) the first player can secure a win of no less than \(B\), no matter how the second player played;
2) the second player can always make it so that the first player will receive no more than \(B\), no matter how he plays.
Two people are playing. The first player writes out numbers from left to right, randomly alternating between 0 and 1, until there are 2021 numbers in total. Each time after the first one writes out the next digit, the second switches two numbers from the already written row (when only one digit is written, the second misses its move). Is the second player always able to ensure that, after his last move, the arrangement of the numbers is symmetrical relative to the middle number?
On a table there are 2022 cards with the numbers 1, 2, 3, ..., 2022. Two players take one card in turn. After all the cards are taken, the winner is the one who has a greater last digit of the sum of the numbers on the cards taken. Find out which of the players can always win regardless of the opponent’s strategy, and also explain how he should go about playing.
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 \(7\times7\) 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 \(x \neq 0\)).
(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.)