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?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

How many different triangles can you make on a circular pegboard that has nine pegs?

These practical challenges are all about making a 'tray' and covering it with paper.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Can you draw a square in which the perimeter is numerically equal to the area?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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

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

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?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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?

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

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?

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

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

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

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

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. . . .

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?

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?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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.

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

Can you find all the different triangles on these peg boards, and find their angles?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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.

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 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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.