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 Queen has introduced a new currency in the world of Wonderland. This currency consists of three golden coins with values \(3\), \(5\) and \(15\). Is it possible for Alice to change an old note with value \(100\) using \(11\) new coins?
One sunny day Alice met the White Rabbit. The Rabbit told her that he owns a pocket watch which has 11 gears arranged in a chain loop. The rabbit asked Alice if it was possible for all the gears to rotate simultaneously. What is your opinion on this matter? Can all the gears rotate simultaneously?
After the Mad Tea-Party, the Hatter was so excited that he decided to cool down by going on a short walk across the chessboard. He started at position a1, then walked around in steps taking each step as if he was a knight, and eventually returned back to a1. Show that he made an even number of steps.
Is it possible that odd integers \(a\), \(b\), \(c\), \(d\) satisfy \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1\)?
The Cheshire Cat wrote one of the numbers \(1, 2,\dots, 15\) into each box of a \(15\times15\) square table in such a way, that boxes which are symmetric to the main diagonal contain equal numbers. Every row and column consists of 15 different numbers. Show that no two numbers along the main diagonal are the same.
The product of 22 integers is equal to 1. Show that their sum cannot be zero.
Is it possible to divide the numbers 1, 2, 3, ..., 100 into pairs of one odd and one even number, such that in every pair except one the even number is greater than the odd number
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”?