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Pack Man

Age 16 to 18 Challenge Level:

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

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Packing

Age 3 to 5

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

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Packing 3D Shapes

Age 14 to 16 Challenge Level:

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

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Packing Boxes

Age 14 to 16 Short Challenge Level:

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.

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Packing Small Boxes

Age 11 to 14 Short Challenge Level:

How many small boxes will fit inside the big box?

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Page Numbers

Age 7 to 11 Short Challenge Level:

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

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Paint Rollers for Frieze Patterns.

Age 11 to 16

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

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Painted Cube

Age 14 to 16 Challenge Level:

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?

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Painted Cube Poster

Age 11 to 14 Challenge Level:

Painted Cube - January 2011

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Painted Faces

Age 7 to 11 Challenge Level:

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?

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Painted Octahedron

Age 11 to 14 Short Challenge Level:

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?

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Painted Purple

Age 14 to 16 Short Challenge Level:

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?

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Painting Between the Lines

Age 11 to 16 Challenge Level:

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?

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Painting by Functions

Age 16 to 18 Challenge Level:

Use functions to create minimalist versions of works of art.

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Painting by Numbers

Age 16 to 18 Challenge Level:

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

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Painting Cubes

Age 11 to 14 Challenge Level:

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?

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Painting Possibilities

Age 7 to 11 Challenge Level:

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. . . .

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Pair Products

Age 14 to 16 Challenge Level:

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

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Pair Products Poster

Age 11 to 14 Challenge Level:

Pair Products Poster - January 2007

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Pair Squares

Age 16 to 18 Challenge Level:

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.

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Pair Sums

Age 11 to 14 Challenge Level:

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

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Paired Parabolas

Age 16 to 18 Challenge Level:

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

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Pairing Up

Age 11 to 14 Short Challenge Level:

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?

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Pairs

Age 11 to 14 Challenge Level:

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. . . .

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Pairs of Legs

Age 5 to 7 Challenge Level:

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

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Pairs of Numbers

Age 5 to 7 Challenge Level:

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?

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Palindromes

Age 5 to 14

Find out about palindromic numbers by reading this article.

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Palindromic Date

Age 7 to 11 Challenge Level:

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

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Palindromic Milometer

Age 11 to 14 Short Challenge Level:

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

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Paper Curves

Age 7 to 11 Challenge Level:

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

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Paper Folding - Models of the Platonic Solids

Age 7 to 16

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

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Paper Halving

Age 5 to 11 Challenge Level:

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

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Paper Partners

Age 5 to 7 Challenge Level:

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

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Paper Patchwork 1

Age 5 to 7 Challenge Level:

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

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Paper Patchwork 2

Age 5 to 7 Challenge Level:

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

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Paper Weight

Age 11 to 14 Short Challenge Level:

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

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Parabella

Age 14 to 18 Challenge Level:

Can you prove an algebraic statement using geometric reasoning?

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Parabella

Age 16 to 18 Challenge Level:

This is a beautiful result involving a parabola and parallels.

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Parabolas Again

Age 14 to 18 Challenge Level:

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

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Parabolic Patterns

Age 14 to 18 Challenge Level:

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.

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Paradoxes

Age 7 to 14

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.

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Parallel Base

Age 11 to 14 Short Challenge Level:

Weekly Problem 46 - 2015
The diagram shows two parallel lines and two angles. What is the value of x?

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Parallel Lines

Age 11 to 14 Challenge Level:

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

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Parallel Parking

Age 14 to 16

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.

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Parallel Universe

Age 14 to 16 Challenge Level:

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

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Parallelo-split

Age 14 to 16 Challenge Level:

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Parallelogram in the Middle

Age 11 to 14 Short Challenge Level:

Weekly Problem 27 - 2013
The diagram shows a parallelogram inside a triangle. What is the value of $x$?

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Parent and Carer Guide

Age 5 to 11

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

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Parent and Carer Guide

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

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Parents, Children and Maths

Age 5 to 7

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