Can you find ways of joining cubes together so that 28 faces are visible?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
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?
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?
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 work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?
Can you fit the tangram pieces into the outlines of the chairs?
Make a flower design using the same shape made out of different sizes of paper.
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!
Can you fit the tangram pieces into the outline of Mai Ling?
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.
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?
Exploring and predicting folding, cutting and punching holes and making spirals.
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
What shape is made when you fold using this crease pattern? Can you make a ring design?
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 fit the tangram pieces into the outlines of these clocks?
Can you fit the tangram pieces into the outlines of the workmen?
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?
Can you fit the tangram pieces into the outline of Little Ming playing the board game?
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?
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 visualise what shape this piece of paper will make when it is folded?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Exchange the positions of the two sets of counters in the least possible number of moves
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Can you fit the tangram pieces into the outlines of these people?
Can you fit the tangram pieces into the outline of Granma T?
Can you cut up a square in the way shown and make the pieces into a triangle?
Can you fit the tangram pieces into the outline of the child walking home from school?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Can you fit the tangram pieces into the outline of Little Fung at the table?
Can you make a 3x3 cube with these shapes made from small cubes?
Here's a simple way to make a Tangram without any measuring or ruling lines.
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?
Which of these dice are right-handed and which are left-handed?
Make a cube out of straws and have a go at this practical challenge.