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.
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
Is it true that if a natural number is divisible by 4 and by 6, then it must be divisible by \(4\times6=24\)?
And what if a natural number is divisible by 5 and by 7? Should it be divisible by 35?
The number \(A\) is not divisible by 3. Is it possible that the number \(2A\) is divisible by 3?
Lisa knows that \(A\) is an even number. But she is not sure if \(3A\) is divisible by 6. What do you think?