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.
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.