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

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?

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 is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

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 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!

How many different symmetrical shapes can you make by shading triangles or squares?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

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

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

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

A huge wheel is rolling past your window. What do you see?

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

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

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

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?

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

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?

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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

Can you fit the tangram pieces into the silhouette of the junk?

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?

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

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

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

Join pentagons together edge to edge. Will they form a ring?

What is the shape of wrapping paper that you would need to completely wrap this model?

Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

Why do you think that the red player chose that particular dot in this game of Seeing Squares?

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

Can you find a way of representing these arrangements of balls?

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

Can you mark 4 points on a flat surface so that there are only two different distances between them?

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

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.