A \(3 \times 3\) magic square is a square with different number from \(1\) to \(9\) in each of its \(9\) cells. The numbers in each row, column and diagonal sum up to \(15\). Show that there is a number \(5\) in the centre of the square.
Can you decorate an \(8 \times 8\) cake with chocolate roses in such a way that any \(2 \times 2\) piece would have exactly 2 roses on it, and any \(3 \times 1\) piece would have exactly one rose? Either draw such a cake or explain why this is not possible.
Several films were nominated for the “Best Math Movie“ award. Each of the 10 judges secretly picked the top movie of their choice. It is known that out of any 4 judges, at least 2 voted for the same film. Prove that there exists a film that was picked by at least 4 judges.
Out of \(7\) integer numbers, the sum of any \(6\) is a multiple of \(5\). Show that every one of these numbers is a multiple of \(5\).
An \(8 \times 8\) chessboard has 30 diagonals total (15 in each direction). Is it possible to place several chess pieces on this chessboard in such a way that the total number of pieces on each diagonal would be odd?
Anna’s garden is a grid of \(n \times m\) squares. She wants to have trees in some of these squares, but she wants the total number of trees in each column and in each row to be an odd number (not necessarily the same, they just all need to be odd). Show that it is possible only if \(m\) and \(n\) are both even or both odd and calculate in how many different ways she can place the trees in the grid.
Anna has \(20\) novels and \(25\) comic books on her shelf. She doesn’t really keep her room very tidy and so she also has a lot of novels and comic books in various places around her room. Each time she reaches for the shelf, she takes two books and puts one back. If she takes two novels or two comic books, she puts a novel back on the shelf. If she takes a novel and a comic book, she places another comic book on the shelf. That way, her shelf sistematically empties. Show that eventually there will be a lone comic book standing on her shelf and all her other books scattered across her room.
A knight in chess moves in an “L” pattern – two squares in one direction and one square in a perpendicular direction. Starting with a knight in the bottom right corner of a regular \(8 \times 8\) chessboard, can you move it some number of times according to the rules in such a way that it visits every square on the chessboard exactly once and ends up in the top left corner?
Nine lightbulbs are arranged in a \(3 \times 3\) square. Some are on, some are off. You are allowed to change the state of all the bulbs in a column or in a row. That means all the bulbs in that row or column that were off light up and the ones that were on go dark. Is it possible to go from the arrangement in the left to the one on the right by repeating this operation?
Anna’s sister, Claire, has \(10\) novels, \(11\) textbooks and \(12\) comic books on her shelf. She also doesn’t like to keep all her books there. Each time she takes two books of different type from the shelf and puts a book of the third type back on. So for example, she might take a novel and a comic book and put a textbook back. Show that eventually there is only a single textbook, and exactly a textbook, left on her shelf.