In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

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?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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

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.

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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.

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?

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, focuses on 'open squares'. What would the next five open squares look like?

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

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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 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?

You have 5 darts and your target score is 44. How many different ways could you score 44?

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?

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?

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

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

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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

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?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

These two group activities use mathematical reasoning - one is numerical, one geometric.

Number problems at primary level to work on with others.

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

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

I was looking at the number plate of a car parked outside. Using my special code S208VBJ adds to 65. Can you crack my code and use it to find out what both of these number plates add up to?

Ben has five coins in his pocket. How much money might he have?

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

Number problems at primary level that may require resilience.

Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Annie cut this numbered cake into 3 pieces with 3 cuts so that the numbers on each piece added to the same total. Where were the cuts and what fraction of the whole cake was each piece?

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

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!

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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. . . .

On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are?

Number problems at primary level that require careful consideration.

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

Use your logical reasoning to work out how many cows and how many sheep there are in each field.