Two friends went simultaneously from A to B. The first went by bicycle, the second – by car at a speed five times faster than the first. Halfway along the route, the car was in an accident, and the rest of the way the motorist walked on foot at a speed half of the speed of the cyclist. Which of them arrived at B first?
Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?
A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?
Show that a knight’s tour is impossible on a \(3\times3\) grid.
Show that two queens together can attack every square on a \(4\times4\) grid, but one queen on her own cannot do it. This type of problem is called ‘queen’s domination’.
How many queens can you place on a \(4\times4\) grid so that none of them attack each other?
Show an knight’s tour on a \(5\times6\) chessboard. That is, a path where a knight starts at one square, and then visits every square exactly once, making only moves legal to a knight.
Show how five queens can dominate a standard \(8\times8\) chessboard. That is, each square is attacked by some queen.
How many independent queens can you place on a \(5\times5\) grid? That is, so none of them attack each other.
How many ways can you place \(8\) rooks independently on a chessboard? That is, so that none of them attack each other.