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

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

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?

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

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?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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?

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?

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

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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?

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?

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?

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

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?

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?

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?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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

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?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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?

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'.

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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. . . .

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

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?

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

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

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

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

Make a cube out of straws and have a go at this practical challenge.

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

What is the greatest number of squares you can make by overlapping three squares?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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?

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

Exploring and predicting folding, cutting and punching holes and making spirals.

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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

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

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