Place coins on a \(6\times 6\) chequered board (one coin on one square), so that all the horizontal lines contain different number of coins, and all vertical lines contain the same number of coins.
Cut a square into a heptagon (7 sides) and an octagon (8 sides) in such a way, that for every side of an octagon there exists an equal side belonging to the heptagon.
Is it possible to arrange some group of distinct numbers in a circle so that each number equals the sum of its neighbours?
A big square was cut into smaller squares. Sebastian used all the pieces and constructed two squares with different side lengths by glueing the pieces together. Show an example of how he could do that.
Sarah is writing down natural numbers starting from 2. She notices that each time she writes the next number the sum of all written numbers is less than their product. She believes she can find such 57 natural numbers (not necessarily different from each other) that their sum will be greater than their product. Do you think it is possible?
(a) Can you find a set of distinct numbers which can be arranged in a circle in such a way that each number equals the product of its neighbours?
(b) Is it true that each solution of Example 1 is determined by the values of two neighbouring numbers?
It was Sebastian’s younger brother who cut the big square in Example 2. Now you need to help him to cut one of the squares (which Sebastian obtained after glueing the pieces) into smaller congruent triangles. But please make sure the elder brother can do the same thing as before: to divide the resulting congruent triangles into two groups and to glue the pieces of each group together to make two squares with different side lengths.
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 do not attack each other. Do you think it is possible or not?
After listening to Harry’s complaints the delivery service promised him to deliver a very expensive chess set together with some books on chess strategies and puzzles. This week one of the tasks was to put 14 bishops on a chessboard so that they do not attack each other. Harry solved this problem and smiled hoping he is not getting 32 identical bishops this time. Can you solve it?
Colour the plane in red, green and blue in such a way that every line consists of points of at most two colours. Remember that you have to use all three colours.