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Problems Methods Algebraic methods Processes and operations Set theory and logic Algorithm Theory Algorithm Theory

A traveller rents a room in an inn for a week and offers the innkeeper a chain of seven silver links as payment – one link per day, with the condition that they will be payed everyday. The innkeeper agrees, with the condition that the traveller can only cut one of the links. How did the traveller manage to pay the innkeeper?

Problem #PRU-88031

Problems Methods Algebraic methods Processes and operations Set theory and logic Algorithm Theory Algorithm Theory

10–13 3.0

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?

Problem #PRU-88044

Problems Set theory and logic Order relations Methods Pigeonhole principle Pigeonhole principle (other)

10–12 2.0

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.

Problem #PRU-88071

Problems Algebra Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Algebraic methods Proof by exhaustion

10–12 2.0

Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.

Problem #PRU-88072

Problems Algebra Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

11–14 2.0

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?

Problem #PRU-88081

Problems Algebra Number systems Decimal number system Methods Algebraic methods Proof by exhaustion

10–12 2.0

Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?

Problem #PRU-88082

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

10–12 2.0

Is it possible to arrange 44 marbles into 9 piles, so that the number of marbles in each pile is different?

Problem #PRU-88103

Problems Methods Pigeonhole principle

10–13 2.0

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?

Problem #PRU-88120

Problems Methods Algebraic methods Counting in two ways Algebra Word Problems Tables and tournaments Number tables and its properties

10–12 2.0

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?

Problem #PRU-88178

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–12 2.0

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?