A look at different crystal lattice structures, and how they relate to structural properties

Some toys have been muddled up! Can you sort them out into boxes? Will they fit in the boxes?

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.

How many small boxes will fit inside the big box?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

What is the smallest number of colours needed to paint the faces of a regular octahedron so that no adjacent faces are the same colour?

Three faces of a $3 \times 3$ cube are painted red, and the other three are painted blue. How many of the 27 smaller cubes have at least one red and at least one blue face?

In abstract and computer generated art, a real object can be represented by a simplified set of lines. Can you create a picture using mathematical instructions?

Use functions to create minimalist versions of works of art.

How many different colours of paint would be needed to paint these pictures by numbers?

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?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

Some parabolas are related to others. How are their equations and graphs connected?

The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?

Ann thought of 5 numbers and told Bob all the sums that could be made by adding the numbers in pairs. The list of sums is 6, 7, 8, 8, 9, 9, 10,10, 11, 12. Help Bob to find out which numbers Ann was. . . .

How many legs do each of these creatures have? How many pairs is that?

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way?

At the beginning and end of Alan's journey, his milometer showed a palindromic number. Can you find his maximum possible average speed?

This is a simple paper-folding activity that gives an intriguing result which you can then investigate further.

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

In how many ways can you halve a piece of A4 paper? How do you know they are halves?

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

How could you use this graph to work out the weight of a single sheet of paper?

Can you prove an algebraic statement using geometric reasoning?

This is a beautiful result involving a parabola and parallels.

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

Weekly Problem 46 - 2015

The diagram shows two parallel lines and two angles. What is the value of x?

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

Scientist Bryan Rickett has a vision of the future - and it is one in which self-parking cars prowl the tarmac plains, hunting down suitable parking spots and manoeuvring elegantly into them.

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Weekly Problem 27 - 2013

The diagram shows a parallelogram inside a triangle. What is the value of $x$?

This page contains information about the NRICH site for parents and carers.

This page is a place to help parents and carers to find NRICH content that might interest them and their children.

Jenny Murray writes about the sessions she leads in schools for parents to work alongside children on mathematical problems, puzzles and games.