Prove that in any group of 6 people there are either three pairs of people who know one another, or three pairs of people who do not know one another.
A warehouse contains 200 boots of each of the sizes 8, 9, and 10. Amongst these 600 boots, 300 are left boots and 300 are right boots. Prove that there are at least 100 usable pairs of boots in the warehouse.
The alphabet of the Ni-Boom-Boom tribe contains 22 consonants and 11 vowels. A word in this language is defined as any combination of letters in which there are no consecutive consonants and no letter is used more than once. The alphabet is divided into 6 non-empty groups. Prove that it is possible to construct a word from all the letters in the group in at least one of the groups.
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?
Imogen’s cat always sneezes before it rains. Today the cat sneezed. “So, it will rain” thinks Imogen. Is she right?