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

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

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?

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

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

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?

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

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?

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

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?

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?

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

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

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 ...

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 visualise what shape this piece of paper will make when it is folded?

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

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?

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 arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

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?

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

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Reasoning about the number of matches needed to build squares that share their sides.

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

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?

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go 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?

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

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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

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

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?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you find a way of counting the spheres in these arrangements?

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

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?

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

You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.

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'.

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Watch this animation. What do you see? Can you explain why this happens?