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?
There are 18 sweets in one piles, and 23 in another. Two play a game: in one go one can eat one pile of sweets, and the other can be divided into two piles. The loser is one who cannot make a move, i.e. before this player’s turn there are two piles of sweets with one sweet in each. Who wins with a regular game?
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 a communication system consisting of 2001 subscribers, each subscriber is connected with exactly \(n\) others. Determine all the possible values of \(n\).
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.
There are two purses and one coin. Inside the first purse is one coin, and inside the second purse is one coin. How can this be?
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?
A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.
Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.