This activity investigates how you might make squares and pentominoes from Polydron.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

Design an arrangement of display boards in the school hall which fits the requirements of different people.

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

How many models can you find which obey these rules?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

What is the best way to shunt these carriages so that each train can continue its journey?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

Can you draw a square in which the perimeter is numerically equal to the area?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten 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?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

An activity making various patterns with 2 x 1 rectangular tiles.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

If you had 36 cubes, what different cuboids could you make?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?