A standard chessboard has more than a quarter of its squares filled with chess pieces. Prove that at least two adjacent squares, either horizontally, vertically, or diagonally, are occupied somewhere on the board.
In how many ways can you rearrange the numbers from 1 to 100 so that the neighbouring numbers differ by no more than 1?
Your task is to find out a five-digit phone number, asking questions that can be answered with either “yes” or “no.” What is the smallest number of questions for which this can be guaranteed (provided that the questions are answered correctly)?
In an \(n\) by \(n\) grid, \(2n\) of the squares are marked. Prove that there will always be a parallelogram whose vertices are the centres of four of the squares somewhere in the grid.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.
A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?
2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.
Father Christmas has an infinite number of sweets. A minute before the New Year, Father Christmas gives some children 100 sweets, while the Snow Maiden takes one sweet from them. Within half a minute before the New Year, Father Christmas gives the children 100 more sweets, and the Snow Maiden again takes one sweet. The same is repeated for 15 seconds, for 7.5 seconds, etc. until the new Year. Prove that the Snow Maiden will be able to take away all the sweets from the children by the New Year.
What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.
The key of the cipher, called the “swivelling grid”, is a stencil made from a square sheet of chequered paper of size \(n \times n\) (where \(n\) is even). Some of the cells are cut out. One side of the stencil is marked. When this stencil is placed onto a blank sheet of paper in four possible ways (marked side up, right, down or left), its cut-outs completely cover the entire area of the square, where each cell is found under the cut-out exactly once. The letters of the message, that have length \(n^2\), are successively written into the cut-outs of the stencil, where the sheet of paper is placed on a blank sheet of paper with the marked side up. After filling in all of the cut-outs of the stencil with the letters of the message, the stencil is placed in the next position, etc. After removing the stencil from the sheet of paper, there is an encrypted message.
Find the number of different keys for an arbitrary even number \(n\).