(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 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?
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) You have a \(10\times20\) chocolate bar and 19 friends. Since you are good at maths they ask you to split this bar into 19 pieces (always breaking along the lines between squares). All the pieces have to be of a rectangular shape. Your friends don’t really care how much they will get, they just want to be special, so you need to split the bar in such way that no two pieces are the same.
(b) The friends are quite impressed by your problem solving skills. But one of them is not that happy with the fact you didn’t get a single piece of the chocolate bar. He thinks you might feel that you are too special, therefore he convinces the others that you should get another \(10\times20\) chocolate bar and now split it into 20 different pieces, all of rectangular shapes (and still you need to break along the lines between squares). Can you do it now?
There are 36 checkers on a \(6\times6\) board (one checker per one cell). They are numbered from 1 to 36. You need to rearrange the checkers so that the ones with numbers which differ by 1 are neighbours (their cells share an edge) and additionally all square numbers have to correspond to the checkers of one (a) horizontal line; (b) of the diagonals. Is it possible or not?
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Can one cut a square into (a) one 30-gon and five pentagons? (b) one 33-gon and three 10-gons?
A young mathematician had quite an odd dream last night. In his dream he was a knight on a \(4\times4\) board. Moreover, he was moving like a knight moves on the usual chessboard. In the morning he could not remember what was actually happening in his dream, though the young mathematician is pretty sure that either
(a) he has passed exactly once through all the cells of the board except for the one at the bottom leftmost corner, or
(b) he has passed exactly once through all the cells of the board.
For each possibility examine if it could happen or not.
Jennifer draws a hexagon, and a line passing through two of its vertices. It turns out one of the figures in which the original hexagon is divided is a heptagon. Show an example of a hexagon and a line for which it is true.