Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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. . . .

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

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 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?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?

Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!

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 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.

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

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?

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

This challenge extends the Plants investigation so now four or more children are involved.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Fill in the missing numbers so that adding each pair of corner numbers gives you the number between them (in the box).

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

A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids.

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!

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?

If you have only four weights, where could you place them in order to balance this equaliser?

Fill in the numbers to make the sum of each row, column and diagonal equal to 34. For an extra challenge try the huge American Flag magic square.

What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.

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?

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

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

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

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

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

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

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

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

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?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

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?

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?

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.

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

A game for 2 players. Practises subtraction or other maths operations knowledge.

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

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!