Can you make a 3x3 cube with these shapes made from small cubes?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Can you find ways of joining cubes together so that 28 faces are visible?

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

Which of these dice are right-handed and which are left-handed?

An activity centred around observations of dots and how we visualise number arrangement patterns.

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

Exchange the positions of the two sets of counters in the least possible number of moves

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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. . . .

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

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?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outline of Little Ming?

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

Make a flower design using the same shape made out of different sizes of paper.

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Can you cut up a square in the way shown and make the pieces into a triangle?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over?

Can you fit the tangram pieces into the outline of these rabbits?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

What shape is made when you fold using this crease pattern? Can you make a ring design?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Can you fit the tangram pieces into the outline of this telephone?

This article for teachers describes a project which explores thepower of storytelling to convey concepts and ideas to children.