How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

This dice train has been made using specific rules. How many different trains can you make?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

Can you make square numbers by adding two prime numbers together?

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Number problems at primary level that require careful consideration.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

Can you use the information to find out which cards I have used?

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

Can you work out some different ways to balance this equation?

What happens when you round these three-digit numbers to the nearest 100?

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you find all the ways to get 15 at the top of this triangle of numbers?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?