If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

Arrange 3 red, 3 blue and 3 yellow counters into a three-by-three square grid, so that there is only one of each colour in every row and every column

Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

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?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

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.

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

How many trains can you make which are the same length as Matt's, using rods that are identical?

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.

Can you find all the different ways of lining up these Cuisenaire rods?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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

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!

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?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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?

An investigation that gives you the opportunity to make and justify predictions.

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?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

Find all the numbers that can be made by adding the dots on two dice.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Can you fill in the empty boxes in the grid with the right shape and colour?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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

Use the numbers and symbols to make this number sentence correct. How many different ways can you find?

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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?

Investigate the different ways you could split up these rooms so that you have double the number.

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

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

Can you find the chosen number from the grid using the clues?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.