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?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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?

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

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

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 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.

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.

A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

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

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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

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?

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?

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?

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?

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

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

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

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?

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

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

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.

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

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

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

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

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?

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. . . .

Can you work out what is wrong with the cogs on a UK 2 pound coin?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

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?

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.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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?

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?

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

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

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

Exchange the positions of the two sets of counters in the least possible number of moves

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Can you fit the tangram pieces into the outlines of the candle and sundial?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Can you fit the tangram pieces into the outlines of these clocks?