ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .

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. . . .

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 cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

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?

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. . . .

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

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?

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

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?

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?

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

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

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.

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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

Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?

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

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

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.

See if you can anticipate successive 'generations' of the two animals shown here.

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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?

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

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

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?

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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. . . .

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

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

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

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

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

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?

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

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

How can you make an angle of 60 degrees by folding a sheet of paper twice?

When dice land edge-up, we usually roll again. But what if we didn't...?

A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?