Problems

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.

The Cheshire Cat wrote one of the numbers $1,2,\dots ,15$ into each box of a $15\times 15$ 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

Look at this formula found by Euler: $n^2+n+41$. It has a remarkable property: for every integer number from $1$ to $21$ it always produces prime numbers. For example, for $n=3$ it is $53$, a prime. For $n=20$ it is $461$, also a prime, and for $n=21$ it is $503$, prime as well. Could it be that this formula produces a prime number for any natural $n$?

Prove that if $a^3-b^3$, for $a$ and $b$ natural, is divisible by $3$, then it is divisible by $9$.

What time is it going to be in $2019$ hours from now?

What is the remainder of $1203\times 1203-1202\times 1205$ when divided by $12$?

Show that a perfect square can only have remainders 0 or 1 when divided by 4.

Convert 2000 seconds into minutes and seconds.