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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Visualising - Upper Secondary

### Alquerque

### Pumpkin Patch

### Introducing NRICH TWILGO

### Seega

### Like a Circle in a Spiral

### Baravelle

### Prime Magic

### Clocking Off

### Cubic Conundrum

### Air Nets

### Rhombus It

### Sprouts

### Hamiltonian Cube

### Funny Factorisation

### Searching for Mean(ing)

### Charting Success

### Diamond Collector

### Charting More Success

### Nine Colours

### Marbles in a Box

### LOGO Challenge - Circles as Animals

### Sliding Puzzle

### Semi-regular Tessellations

### Shaping the Universe I - Planet Earth

### Cuboid Challenge

### LOGO Challenge - Triangles-squares-stars

### Shaping the Universe II - the Solar System

### Shaping the Universe III - to Infinity and Beyond

### What Is the Question?

### Parallelogram It

### Yih or Luk Tsut K'i or Three Men's Morris

### Ding Dong Bell

### The Bridges of Konigsberg

### Triangles in the Middle

### Instant Insanity

### Just Rolling Round

### Picture Story

### Doesn't Add Up

### One and Three

### Corridors

### Bike Shop

### Slippage

### Speeding Boats

### Mystic Rose

### Centre Square

### Factorising with Multilink

### Coke Machine

### Proximity

### Fermat's Poser

### Concrete Calculation

### Changing Places

### Bus Stop

### Twelve Cubed

### Packing 3D Shapes

### Painted Purple

### Folding in Half

### Semicircular Design

### Just Opposite

### Natural Sum

### A Problem of Time

### Painted Cube

### Inside Out

### Triangles Within Triangles

### Rectangle Rearrangement

### Dicey Directions

### Partly Painted Cube

### Packing Boxes

### Relative Time

### Travelling by Train

### Building Tetrahedra

### Hypotenuse Lattice Points

### Tilting Triangles

### Jam

### Tetrahedra Tester

### Sliced

### Triangles Within Squares

### Spotting the Loophole

### Newspaper Sheets

### Eulerian

### Oldest and Youngest

### Platonic Planet

### Star Gazing

### Around and Back

### A Tilted Square

### Wari

### Circuit Training

### Triangles Within Pentagons

### Making Tracks

### A Question of Scale

### Summing Squares

### Pyramidal N-gon

### Travelator

### Penta Colour

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Age 5 to 18

This game for two, was played in ancient Egypt as far back as 1400 BC. The game was taken by the Moors to Spain, where it is mentioned in 13th century manuscripts, and the Spanish name Alquerque derives from the Arabic El- quirkat. Watch out for being 'huffed'.

Age 5 to 18

A game for two players based on a game from the Somali people of Africa. The first player to pick all the other's 'pumpkins' is the winner.

Age 5 to 18

Challenge Level

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

Age 5 to 18

An ancient game for two from Egypt. You'll need twelve distinctive 'stones' each to play. You could chalk out the board on the ground - do ask permission first.

Age 7 to 16

Challenge Level

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

Age 7 to 16

Challenge Level

What can you see? What do you notice? What questions can you ask?

Age 7 to 16

Challenge Level

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

Age 7 to 16

Challenge Level

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

Age 7 to 16

Challenge Level

Which of the following cubes can be made from these nets?

Age 7 to 18

Challenge Level

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Age 11 to 16

Challenge Level

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 rhombus.

Age 11 to 16

Challenge Level

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

Age 11 to 16

ShortChallenge Level

Weekly Problem 36 - 2007

Find the length along the shortest path passing through certain points on the cube.

Age 11 to 16

Challenge Level

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

Age 11 to 16

Challenge Level

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

Age 11 to 16

Challenge Level

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Age 11 to 16

Challenge Level

Collect as many diamonds as you can by drawing three straight lines.

Age 11 to 16

Challenge Level

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Age 11 to 16

Challenge Level

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Age 11 to 16

Challenge Level

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

Age 11 to 16

Challenge Level

See if you can anticipate successive 'generations' of the two animals shown here.

Age 11 to 16

Challenge Level

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.

Age 11 to 16

Challenge Level

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

Age 11 to 16

This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.

Age 11 to 16

Challenge Level

What's the largest volume of box you can make from a square of paper?

Age 11 to 16

Challenge Level

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.

Age 11 to 16

The second in a series of articles on visualising and modelling shapes in the history of astronomy.

Age 11 to 16

The third installment in our series on the shape of astronomical systems, this article explores galaxies and the universe beyond our solar system.

Age 11 to 16

Challenge Level

These pictures and answers leave the viewer with the problem "What is the Question". Can you give the question and how the answer follows?

Age 11 to 16

Challenge Level

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 parallelogram.

Age 11 to 18

Challenge Level

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and knot arithmetic.

Age 11 to 18

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

Age 11 to 18

Challenge Level

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

Age 11 to 18

Challenge Level

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Age 11 to 18

Challenge Level

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Age 14 to 16

Challenge Level

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Age 14 to 16

Challenge Level

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Age 14 to 16

Challenge Level

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Age 14 to 16

Challenge Level

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400 metres from B. How long is the lake?

Age 14 to 16

Challenge Level

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

Age 14 to 16

ShortChallenge Level

If I walk to the bike shop, but then cycle back, what is my average speed?

Age 14 to 16

Challenge Level

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance up the wall which the ladder can reach?

Age 14 to 16

Challenge Level

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Age 14 to 16

Challenge Level

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Age 14 to 16

ShortChallenge Level

What does Pythagoras' Theorem tell you about the radius of these circles?

Age 14 to 16

Challenge Level

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

Age 14 to 16

Challenge Level

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

Age 14 to 16

Challenge Level

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Age 14 to 16

Challenge Level

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

Age 14 to 16

Challenge Level

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations?

Age 14 to 16

Challenge Level

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves it will take to move the red counter to HOME?

Age 14 to 16

Challenge Level

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant and in the ratio 5 to 4. The buses travel to and fro between the towns. What milestones are at Shipton and Veston?

Age 14 to 16

ShortChallenge Level

A wooden cube with edges of length 12cm is cut into cubes with edges of length 1cm. What is the total length of the all the edges of these centimetre cubes?

Age 14 to 16

Challenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

How does the perimeter change when we fold this isosceles triangle in half?

Age 14 to 16

ShortChallenge Level

Weekly Problem 9 - 2016

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

Age 14 to 16

Challenge Level

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

Age 14 to 16

Challenge Level

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural numbers.

Age 14 to 16

Challenge Level

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

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?

Age 14 to 16

Challenge Level

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you can colour every face of all of the smaller cubes?

Age 14 to 16

Challenge Level

Can you find a rule which connects consecutive triangular numbers?

Age 14 to 16

ShortChallenge Level

A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

Age 14 to 16

ShortChallenge Level

An ordinary die is placed on a horizontal table with the '1' face facing East... In which direction is the '1' face facing after this sequence of moves?

Age 14 to 16

Challenge Level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Age 14 to 16

ShortChallenge 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.

Age 14 to 16

ShortChallenge Level

Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?

Age 14 to 16

ShortChallenge Level

Stephen stops at Darlington on his way to Durham. At what time does he arrive at Durham?

Age 14 to 16

Challenge Level

Can you make a tetrahedron whose faces all have the same perimeter?

Age 14 to 16

Challenge Level

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

Age 14 to 16

Challenge Level

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

Age 14 to 16

Challenge Level

To avoid losing think of another very well known game where the patterns of play are similar.

Age 14 to 16

Challenge Level

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?

Age 14 to 16

Challenge Level

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Age 14 to 16

Challenge Level

Can you find a rule which relates triangular numbers to square numbers?

Age 14 to 16

Challenge Level

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

Age 14 to 16

ShortChallenge Level

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

Age 14 to 16

ShortChallenge Level

Weekly Problem 37 - 2014

Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

Age 14 to 16

ShortChallenge Level

Edith had 9 children at 15 month intervals. If the oldest is now six times as old as the youngest, how old is her youngest child?

Age 14 to 16

Challenge Level

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

Age 14 to 16

Challenge Level

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Age 14 to 16

Challenge Level

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns around and heads back to the starting point where he meets the runner who is just finishing his first circuit. Find the ratio of their speeds.

Age 14 to 16

Challenge Level

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Age 14 to 16

Challenge Level

This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

Age 14 to 16

Challenge Level

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever meet at the start again? If so, after how many circuits?

Age 14 to 16

Challenge Level

Show that all pentagonal numbers are one third of a triangular number.

Age 14 to 16

Challenge Level

A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel?

Age 14 to 16

Challenge Level

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

Age 14 to 16

Challenge Level

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Age 14 to 16

ShortChallenge Level

The base of a pyramid has n edges. What is the difference between the number of edges the pyramid has and the number of faces the pyramid has?

Age 14 to 16

ShortChallenge Level

When Andrew arrives at the end of the walkway, how far is he ahead of Bill?

Age 14 to 16

Challenge Level

In how many different ways can I colour the five edges of a pentagon so that no two adjacent edges are the same colour?