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

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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?

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. . . .

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?

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

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

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?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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

Can you explain why it is impossible to construct this triangle?

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

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. . . .

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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 game for 2 people. Take turns joining two dots, until your opponent is unable to move.

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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.

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. . . .

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. . . .

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?

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

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.

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?

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.

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

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . .

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

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?

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

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

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?

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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

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?

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

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. . . .