On the first day of school, in all three of the first year classes (1A, 1B, 1C), there were three lessons: Maths, French and Biology. Two classes cannot have the same lesson at the same time. 1B’s first lesson was Maths. The Biology teacher praised the students in 1B: “You have even better marks than 1A”. 1A’s second lesson was not French. Which class’s last lesson was Biology?
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 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?
The board has the form of a cross, which is obtained if corner boxes of a square board of \(4 \times 4\) are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?
Prove that: \[a_1 a_2 a_3 \cdots a_{n-1}a_n \times 10^3 \equiv a_{n-1} a_n \times 10^3 \pmod4,\] where \(n\) is a natural number and \(a_i\) for \(i=1,2,\ldots, n\) are the digits of some number.