The date 21.02.2012 reads the same forwards and backwords (such numbers are called palindromes). Are there any more palindrome dates in the twenty first centuary?
Do there exist three natural numbers such that neither of them divide each other, but each number divides the product of the other two?
Find all the solutions of the puzzle and prove there are no others. Different letters denote different digits, while the same letters correspond to the same digits. \[M+MEEE=BOOO.\]
Is it possible to arrange some group of distinct numbers in a circle so that each number equals the sum of its neighbours?
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’s not that difficult to find a set of \(57\) integers which has a product strictly larger or strictly smaller than their sum. Is it possible to find \(57\) integers (not necessarily distinct) with their sum being equal to their product?
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?
Bella spent 10 minutes searching for a 3-digit number such that it has the product of it’s digits equal to 26. She examined all 3-digit numbers one by one. Do you think she missed a possible example or is it the case that there are simply no such 3-digit numbers?