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.
Cut a rectangle into two identical pentagons.
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) Cut the rectangle into two identical quadrilaterals.
(b) Cut the rectangle into two identical hexagons.
(c) Cut the rectangle into two identical heptagons.
(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?
Three liberals and three democrats are about to cross the river. The only available boat has two seats. The difficult part is that the democrats do not want to be outnumbered on any of the shores while the boat is on water (they are afraid in such case they will get pushed into the river by the liberals). To avoid another civil war you need to organise their transit properly.