Problems

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse (from the node “M”) moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.

Can the cat catch the mouse in each of the cases A, B, C?

Two players in turn increase a natural number in such a way that at each increase the difference between the new and old values of the number is greater than zero, but less than the old value. The initial value of the number is 2. The winner is the one who can create the number 1987. Who wins with the correct strategy: the first player or his partner?

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.

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?

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?