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.
There are six cities in Wonderland. Her Majesty’s principal secretary of state for transport has a plan of building six new railways. The only condition for these railways is that each of them joins some pair of cities having other four cities equally distributed on both sides of a line containing the segment of the railway. Is it possible to implement such a plan for some configuration of cities?
Ten ladies and ten gentlemen regularly attend a dancing club. Last week the participants gave a short performance for their relatives and friends. They showed ten different dances. Every gentleman had a chance to dance once with every lady. It turns out that every lady danced her next dance with either blonder or taller partner than the previous dance. Explain how that could be possible.
More problems about chessboard and chess pieces:
(a) Can it be true that there are only 8 knights on a \(4\times12\) board and each empty cell is attacked by at least one of the knights?
(b) Put some number of knights on a chessboard in such a way that each knight attacks exactly three other knights.
Draw 16 diagonals inside some cells of a \(5\times5\) square in such a way that no two of these diagonals share any points.
This academic year Harry decided not only to attend Maths Circles, but also to join his local Chess Club. Harry’s chess set was very old and some pieces were missing, so he ordered a new one. When it arrived, he found out to his surprise that the set consisted of 32 knights of different colours. He was a bit upset but he decided to spend some time on solving the problem he heard on the last Saturday’s Maths Circle session. The task was to find out if it is possible to put more than 30 knights on a chessboard in such a way that they cannot attack each other. Do you think it is possible or not?
(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 the distance of 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 exactly once? (The robot is not allowed to leave the strip.)
(b) What if the task is to visit exactly once all cells of a \(1\times99\) strip? (The robot is not allowed to leave the strip.)
Ten ladies and ten gentlemen regularly attend a dancing club. Last week the participants gave a short performance for their relatives and friends. They showed ten different dances. Every gentleman had a chance to dance once with every lady. It turns out that every lady danced her next dance with either blonder or taller partner than the previous dance. Explain how that could be possible.