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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.

On the way back from his weekly maths circle Harry created the following puzzle:

Put 48 rooks on a \(10\times10\) board so that each rook attacks only 2 or 4 empty cells.

When he showed this problem to the teachers next Saturday they were very impressed and decided to include it in the next problem set. Try to find a suitable placement of rooks.

Can you arrange numbers from 1 to 9 in one line so that sums of digits of neighbouring numbers differ only by 2 or by 3?

(a) Can you do the same trick (see Example 2) with numbers from 1 to 17?

(b) Can you do it with numbers from 1 to 19?

(c) Can one arrange them (numbers from 1 to 19) in a circle with the same condition being satisfied?

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?

There are six cities in Wonderland. Her Majesty’s principal secretary of state for transport has a plan of building six new railways. The only condition for these railways is that each of them joins some pair of cities having other four cities equally distributed on both sides of a line containing the segment of the railway. Is it possible to implement such a plan for some configuration of cities?

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.