You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.
Is it possible to place the numbers \(-1, 0, 1\) in a \(6\times 6\) square such that the sums of each row, column, and diagonal are unique?
There are bacteria in a glass. After a second each bacterium divides in half to create two new bacteria. Then after another second these bacteria divide in half, and so on. After a minute the glass is full. After how much time will the glass be half full?
Anna, Vincent, Tom and Sarah each bought one apple for 10p from a fruit stand. How did they manage to do this, if they didn’t have any coins less than 20p and if the fruit stand didn’t have any change less than 50p?
A piece fell out of a book, the first page of which is the number 439, and the number of the last page is written with those same numbers in some other order. How many pages are in the fallen out piece?
A snail crawls along a wall, having started from the bottom of the wall. Each day the snail crawls upwards by 5 cm and each night it slides down the wall by 4 cm. When does it reach the top of the wall, if the height of the wall is 75 cm?
In January of a certain year there were four Fridays and four Mondays. Which day of the week was the 20th of January in that year?
A rectangle of size \(199\times991\) is drawn on squared paper. How many squares intersect the diagonal of the rectangle?
The intelligence agency of the Galactic Empire intercepted the following coded message from the enemy planet Medusa: \(ABCDE+BADC=ACDED\).
It is known that different numbers are represented by different letters, and that the same numbers are represented by the same letters. Two robots attempted to decode this message and each one got a different answer. Is this possible, or should one of the robots be melted down as scrap metal?
Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?