A battalion of soldiers was marching towards a captured city. Their progress was stopped by a wide river. Fortunately, close to the shore there were two boys sailing in a small boat. They escaped from the city and were eager to help the soldiers to cross the river. The only obstacle was that their boat could fit either two boys or one soldier. Taking into account one person was enough to handle that kind of boat (i.e. to sail from one shore to another) and the fact that on the next day the city was liberated so the boys could reunite with their families describe how the battalion was capable of crossing the river.
The cube from Example 3 is a present and one layer of a gift-wrap is totally not enough. Can you cover it with another 15 identical rectangles? You can assume the covering from Example 3 was thin and it did not affect the shape of a cube. As before no overlappings are allowed and the surface has to be fully covered by rectangles.
(a) Jimmy is working on a metal model of a grasshopper. He named it Kimmy. The boy keeps on adding new features to his robot. Besides being an accurate alarm clock, Kimmy can jump in one or two cells depending on how many times Jimmy claps his hands. Do you think the boy can choose a sequence of claps in such a way that the robot will visit all cells of a \(1\times101\) strip?
(b) What if the task is to visit all cells of a \(1\times99\) strip?
Two clowns A and B are playing the following game. They have 33 tomatoes on a plate. One of the tomatoes is rotten and both clowns know which one. Each move they can choose one, two, or three of the remaining tomatoes from the plate and smash them into their own faces. They take turns and the clown who chooses the rotten tomato looses the game. They cannot skip the moves. Clown A starts the game. Does A or B have a winning strategy? (A winning strategy is a strategy following which you win no matter how your opponent plays.)
On the way back from his weekly maths circle Harry created the following puzzle:
Put 48 rooks on a \(10\times10\) board so that each rook attacks only 2 or 4 empty cells.
When he showed this problem to the teachers next Saturday they were very impressed and decided to include it in the next problem set. Try to find a suitable placement of rooks.
Can you arrange numbers from 1 to 9 in one line so that sums of digits of neighbouring numbers differ only by 2 or by 3?
There are 30 people standing in a queue in front of a candy shop. We know that among every ten people standing one after another there are more adults than kids. Is it possible that among all 30 people there are more kids than adults?
(a) Can you do the same trick (see Example 2) with numbers from 1 to 17?
(b) Can you do it with numbers from 1 to 19?
(c) Can one arrange them (numbers from 1 to 19) in a circle with the same condition being satisfied?
Half an hour later (see example 3) there are only 15 people standing in the queue. The condition about every ten consecutive members of the queue is still satisfied. Can we guarantee that there are more adults than kids?
A boy is playing on a \(4\times10\) board. He is trying to put 8 bishops on the board so that each cell is attacked by one of the bishops. Finally he manages to solve this problem.
(a) Can you show a possible solution?
(b) Can you do the same thing with 7 bishops?