Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

Build a scaffold out of drinking-straws to support a cup of water

Can Jo make a gym bag for her trainers from the piece of fabric she has?

This package contains hands-on code breaking activities based on the Enigma Schools Project. Suitable for Stages 2, 3 and 4.

What shape would fit your pens and pencils best? How can you make it?

These models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

As part of Liverpool08 European Capital of Culture there were a huge number of events and displays. One of the art installations was called "Turning the Place Over". Can you find our how it works?

Make some celtic knot patterns using tiling techniques

What shape and size of drinks mat is best for flipping and catching?

Did you know mazes tell stories? Find out more about mazes and make one of your own.

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them.

This article for students gives some instructions about how to make some different braids.

What happens when a procedure calls itself?

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

What do these two triangles have in common? How are they related?

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

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

This part introduces the use of Logo for number work. Learn how to use Logo to generate sequences of numbers.

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

Exploring and predicting folding, cutting and punching holes and making spirals.

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

Turn through bigger angles and draw stars with Logo.

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

Write a Logo program, putting in variables, and see the effect when you change the variables.

Learn about Pen Up and Pen Down in Logo

More Logo for beginners. Now learn more about the REPEAT command.

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

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

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?

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

A description of how to make the five Platonic solids out of paper.

If you'd like to know more about Primary Maths Masterclasses, this is the package to read! Find out about current groups in your region or how to set up your own.

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

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?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Learn to write procedures and build them into Logo programs. Learn to use variables.

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.

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's a simple way to make a Tangram without any measuring or ruling lines.

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.