100 fare evaders want to take a train, consisting of 12 coaches, from the first to the 76th station. They know that at the first station two ticket inspectors will board two coaches. After the 4th station, in the time between each station, one of the ticket inspectors will cross to a neighbouring coach. The ticket inspectors take turns to do this. A fare evader can see a ticket inspector only if the ticket inspector is in the next coach or the next but one coach. At each station each fare evader has time to run along the platform the length of no more than three coaches – for example at a station a fare evader in the 7th coach can run to any coach between the 4th and 10th inclusive and board it. What is the largest number of fare evaders that can travel their entire journey without ever ending up in the same coach as one of the ticket inspectors, no matter how the ticket inspectors choose to move? The fare evaders have no information about the ticket inspectors beyond that which is given here, and they agree their strategy before boarding.
Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence \(1, 2, \dots , 100, 101\). After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.
27 coins are given, of which one is a fake, and it is known that a counterfeit coin is lighter than a real one. How can the counterfeit coin be found from 3 weighings on the scales without weights?
On a board of size \(8 \times 8\), two in turn colour the cells so that there are no corners of three coloured squares. The player who can’t make a move loses. Who wins with the right strategy?
On a plane there are 100 sheep-points and one wolf-point. In one move, the wolf moves by no more than 1, after which one of the sheep moves by a distance of no more than 1, after that the wolf again moves, etc. At any initial location of the points, will a wolf be able to catch one of the sheep?
A \(99 \times 99\) chequered table is given, each cell of which is painted black or white. It is allowed (at the same time) to repaint all of the cells of a certain column or row in the colour of the majority of cells in that row or column. Is it always possible to have that all of the cells in the table are painted in the same colour?
10 guests came to a party and each left a pair of shoes in the corridor (all guests have the same shoes). All pairs of shoes are of different sizes. The guests began to disperse one by one, putting on any pair of shoes that they could fit into (that is, each guest could wear a pair of shoes no smaller than his own). At some point, it was discovered that none of the remaining guests could find a pair of shoes so that they could leave. What was the maximum number of remaining guests?
How can one measure out 15 minutes, using an hourglass of 7 minutes and 11 minutes?
Two boys play the following game: they take turns placing rooks on a chessboard. The one who wins is the one whose last move leaves all the board cells filled. Who wins if both try to play with the best possible strategy?
In a basket there are 13 apples. There are scales, with which you can find out the total weight of any two apples. Think of a way to find out from 8 weighings the total weight of all the apples.