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.
After having a very “spiritual” conversation with her personal yoga instructor Mrs. Robinson decided to learn more about Feng shui. In her guest room there are 16 chairs and she never liked the way they are standing. She has read through some books on Feng shui and searched through the internet looking for a solution to her problem. Finally, Mrs. Robinson found a short paragraph in one of the old issues of a national magazine about health and home. In this paragraph it was explained that one has to rearrange chairs in such a way that there are exactly 5 chairs standing next to each wall. Taking into account Mrs. Robinson’s guest room has a shape of a square describe how she needs to rearrange the chairs in order to satisfy the condition. (Feng shui is a Chinese philosophical system of harmonizing everyone with the surrounding environment.
Six glasses are placed in a row. The leftmost three are full, and the other three are empty. By doing manipulations with just one glass you need to make empty glasses alternate with full glasses.
In the following puzzle an example on multiplication is encrypted with the letters of Latin alphabet: \[{BAN}\times {G}= {BOOO}.\] Different letters correspond to different digits, identical letters correspond to identical digits. The task is to solve the puzzle.
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?
a) Joker prepares 13 blank cards. He writes a natural number on each of them. (Natural numbers are whole positive numbers.) Then for all 13 numbers he calculates their product and sum. Joker gets the same result for both. Is this some kind of trick or is it really possible? Why?
(b) What is the answer if we don’t know how many cards he uses but we know that both results are equal to 13?
The edges of a cube are assigned with integer values. For each vertex we look at the numbers corresponding to the three edges coming from this vertex and add them up. In case we get 8 equal results we call such cube “cute”. Are there any “cute” cubes with the following numbers corresponding to the edges:
(a) \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\);
(b) \(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6\)?
A candle that burns out completely in one hour costs \(60\) pence and a candle that lasts for \(11\) minutes costs \(11\) pence. For some reasons which he doesn’t want to explain to the seller, Mr. Fawkes wants to measure exactly \(1\) minute with some of these candles. Can he manage to do that if he has only \(1\) pound and \(50\) pence to spend on the candles?
Mr. Fawkes can only measure the whole interval with each of the candles. He is not guaranteed that any of the candles burns uniformly, so he cannot divide it into two parts. He cannot make it burn faster by igniting them from both ends, but he can extinguish a candle and light it up again later.