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) 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 product of 22 integers is equal to 1. Show that their sum cannot be zero.
Back in the days when a young mathematician was even younger he could only draw digits “4” and “7”. While looking through the old notes his mother found one piece of paper on which he wrote the numbers with digit sums equal to 18, 22 and 26. Which numbers could be written on this piece of paper?
a) It seems that the young mathematician was making progress quite fast. On the back side of that piece of paper there are numbers with digits adding up to all natural numbers from 18 to 33. And yet all of them consist of only digits “4” and “7”. Make your own list of that kind.
(b) Is it true that any natural number greater than 17 can be equal to the digit sum of some number written with digits “4” and “7”?
(c) Now let’s try the same question for digits “5” and “8”. What values can you get if you consider the sum of the digits of a number written with the help of digits “5” and “8”?
Divide 15 walnuts into four groups, each group consisting of a different whole number of nuts.
In the following example with fractions replace “stars” with different natural numbers in order to obtain an identity: \[\frac{1}{*}+\frac{1}{*}=\frac{1}{*}+\frac{1}{*}.\]
Looking back at Example 12.1 what if we additionally require all differences to be less than the smallest of the three numbers?
(a) Divide 55 walnuts into four groups consisting of different number of nuts.
(b) Divide 999 walnuts into four groups consisting of different number of nuts.