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

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

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

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 ...

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?

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

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

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

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

How much of the square is coloured blue? How will the pattern continue?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

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?

What can you see? What do you notice? What questions can you ask?

What's the largest volume of box you can make from a square of paper?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

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?

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

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

Can you describe this route to infinity? Where will the arrows take you next?

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.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

If you move the tiles around, can you make squares with different coloured edges?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Here are shadows of some 3D shapes. What shapes could have made them?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?