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?
Notice that the square number 1089 \((=33^2)\) has two even and two odd digits in its decimal representation.
(a) Can you find a 6-digit square number with the same property (the number of odd digits equals the number of even digits)?
(b) What about such 100-digit square number?
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?
Assume you have a chance to play the following game. You need to put numbers in all cells of a \(10\times10\) table so that the sum of numbers in each column is positive and the sum of numbers in each row is negative. Once you put your numbers you cannot change them. You need to pay £1 if you want to play the game and the prize for completing the task is £100. Is it possible to win?
After proving there are no 3-digit numbers with the product of digits equal to 26 (see Example 1) Bella decided to find a 4-digit number with the product of digits equal to 98. Can she succeed in finding such a number?
Bella was encouraged by the fact that she fully understood the general concept about the existence of a number with given value of product of digits. Therefore, she started thinking about the following problems:
(a) Is there a 3-digit number with the sum of digits equal to 24?
(b) Is there a 4-digit number with the sum of digits equal to 37?
Solve these questions.
Once again consider the game from Example 2.
(a) Will you change your answer if the field is a rectangle?
(b) The rules are changed. Now you win if the sum of numbers in each row is greater than 100 and the sum of the numbers in each column is less than 100. Is it possible to win?
Express the number 111 as a sum of 51 natural numbers so that each of the terms has the same sum of digits.
a) Express the number 221 as a sum of 52 natural numbers so that each of the terms has the same sum of digits.
(b) Express the number 226 as a sum of 52 natural numbers so that all terms have the same sum of digits.