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?
Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.
A cube with side length of 20 is divided into 8000 unit cubes, and on each cube a number is written. It is known that in each column of 20 cubes parallel to the edge of the cube, the sum of the numbers is equal to 1 (the columns in all three directions are considered). On some cubes a number 10 is written. Through this cube there are three layers of \(1 \times 20 \times 20\) cubes, parallel to the faces of the cube. Find the sum of all the numbers outside of these layers.
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?
In Conrad’s collection there are four royal gold five-pound coins. Conrad was told that some two of them were fake. Conrad wants to check (prove or disprove) that among the coins there are exactly two fake ones. Will he be able to do this with the help of two weighings on weighing scales without weights? (Counterfeit coins are the same in weight, real ones are also the same in weight, but false ones are lighter than real ones.)
Janine and Zahara each thought of a natural number and said them to Alex. Alex wrote the sum of the thought of numbers onto one sheet of paper, and on the other – their product, after which one of the sheets was hidden, and the other (on it was written the number of 2002) was shown to Janine and Zahara. Seeing this number, Janine said that she did not know what number Zahara had thought of. Hearing this, Zahara said that she did not know what number Janine had thought of. What was the number which Zahara had thought of?
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 case of Brown, Jones and Smith is being considered. One of them committed a crime. During the investigation, each of them made two statements. Brown: “I did not do it. Jones did not do it. " Smith: “I did not do it. Brown did it. “Jones:" Brown did not do it. This was done by Smith. “Then it turned out that one of them had told the truth in both statements, another had lied both times, and the third had told the truth once, and he had lied once. Who committed the crime?