There are 6 locked suitcases and 6 keys for them. It is not known which keys are for which suitcase. What is the smallest number of attempts do you need in order to open all the suitcases? How many attempts would you need if there are 10 suitcases and keys instead of 6?
Four friends came to an ice-rink, each with her brother. They broke up into pairs and started skating. It turned out that in each pair the “gentleman” was taller than the “lady” and no one is skating with his sister. The tallest boy in the group was Sam Smith, Peter Potter, then Luisa Potter, Joe Simpson, Laura Simpson, Dan Caldwell, Jane Caldwell and Hannah Smith. Determine who skated with whom.
Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.
a) Prove that within any 6 whole numbers there will be two that have a difference between them that is a multiple of 5.
b) Will this statement remain true if instead of the difference we considered the total?
Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?
Is it possible to arrange 44 marbles into 9 piles, so that the number of marbles in each pile is different?
In a room, there are 85 red and blue balloons. It is known that: 1) at least one of the balloons is red; 2) from each arbitrarily chosen pair of balloons at least one blue. How many red balloons are there in the room?
Is it possible to fill a \(5 \times 5\) table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?
Is it always the case that in any 25 GBP banknotes – that is £5, £10, £20, and £50 notes – there will always be 7 banknotes of the same denomination?
A class contains 38 pupils. Prove that within the class there will be at least 4 pupils born in the same month.