Visualising - Upper Secondary

interactivity

Introducing NRICH TWILGO

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?

game

Pumpkin patch

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.

game

Seega

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.

game

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'.

problem

Air 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.

problem

Prime magic

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?

problem

Like a circle in a spiral

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?

problem

Cubic conundrum

Age

7 to 16

Challenge level

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

problem

Fruity totals

Age

7 to 16

Challenge level

In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?

problem

Clocking off

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?

game

Favourite

Vector racer

Age

11 to 16

Challenge level

The classic vector racing game.

problem

LOGO challenge - triangles-squares-stars

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.

problem

Charlie's delightful machine

Age

11 to 16

Challenge level

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

problem

Triangle in a trapezium

Age

11 to 16

Challenge level

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

problem

Searching for mean(ing)

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?

article

Ding dong bell

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.

problem

Take three from five

Age

11 to 16

Challenge level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

problem

Cuboid challenge

Age

11 to 16

Challenge level

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

game

Yih or Luk tsut k'i or Three Men's Morris

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.

problem

Wipeout

Age

11 to 16

Challenge level

Can you do a little mathematical detective work to figure out which number has been wiped out?

article

Shaping the universe I - planet Earth

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.

problem

Charting success

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?

game

Sprouts

Age

11 to 16

Challenge level

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

article

Shaping the universe II - the solar system

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

problem

Who's who?

Age

11 to 16

Challenge level

Can you solve the clues to find out who's who on the friendship graph?

problem

Charting more success

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?

article

Shaping the universe III - to infinity and beyond

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

problem

Marbles in a box

Age

11 to 16

Challenge level

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

problem

What is the question?

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?

problem

11x11 square

Age

11 to 16

Challenge level

Here's a neat trick you can do with an 11 by 11 square...

problem

Spots and measles

Age

11 to 16

Challenge level

99% of people who have measles have spots. Ben has spots. Do you think he has measles?

game

The triangle game

Age

11 to 16

Challenge level

Can you discover whether this is a fair game?

problem

Instant insanity

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.

problem

Hamiltonian cube

Age

11 to 16

Challenge level

Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.

problem

Multiple surprises

Age

11 to 16

Challenge level

Sequences of multiples keep cropping up...

problem

Funny factorisation

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?

problem

Triangles in the middle

Age

11 to 18

Challenge level

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

problem

Nine colours

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?

problem

What's it worth?

Age

11 to 16

Challenge level

There are lots of different methods to find out what the shapes are worth - how many can you find?

problem

The bridges of Konigsberg

Age

11 to 18

Challenge level

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

game

Sliding puzzle

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.

problem

LOGO challenge - circles as animals

Age

11 to 16

Challenge level

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

problem

Pythagoras proofs

Age

11 to 16

Challenge level

Can you make sense of these three proofs of Pythagoras' Theorem?

problem

Bent out of shape

Age

14 to 18

Challenge level

An introduction to bond angle geometry.

problem

Gnomon dimensions

Age

14 to 16

Challenge level

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

problem

Dicey directions

Age

14 to 16

Challenge 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?

problem

Relative time

Age

14 to 16

Challenge level

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

problem

Vector walk

Age

14 to 18

Challenge level

Starting with two basic vector steps, which destinations can you reach on a vector walk?

problem

Isosceles seven

Age

14 to 16

Challenge level

Is it possible to find the angles in this rather special isosceles triangle?

problem

The spider and the fly

Age

14 to 16

Challenge level

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

problem

Fermat's poser

Age

14 to 16

Challenge level

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

problem

Triangles within triangles

Age

14 to 16

Challenge level

Can you find a rule which connects consecutive triangular numbers?

problem

Vector journeys

Age

14 to 18

Challenge level

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

problem

Partly circles

Age

14 to 16

Challenge level

What is the same and what is different about these circle questions? What connections can you make?

problem

Eulerian

Age

14 to 16

Challenge 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?

problem

Terminology

Age

14 to 16

Challenge level

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

problem

All tied up

Age

14 to 16

Challenge level

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

problem

A problem of time

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?

problem

Triangles within squares

Age

14 to 16

Challenge level

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

problem

Factorising with multilink

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?

problem

Iff

Age

14 to 18

Challenge level

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

problem

Pyramidal n-gon

Age

14 to 16

Challenge 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?

problem

Summing squares

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?

problem

Semicircular design

Age

14 to 16

Challenge 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?

problem

Bendy quad

Age

14 to 16

Challenge level

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

problem

Curve hunter

Age

14 to 18

Challenge level

This problem challenges you to sketch curves with different properties.

problem

Hypotenuse lattice points

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?

problem

Triangles within pentagons

Age

14 to 16

Challenge level

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

problem

Fill me up too

Age

14 to 16

Challenge level

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

problem

Tetra square

Age

14 to 18

Challenge level

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

problem

A question of scale

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?

problem

In or out?

Age

14 to 16

Challenge level

Weekly Problem 52 - 2014
Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

problem

Pick's theorem

Age

14 to 16

Challenge level

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

problem

Travelling by train

Age

14 to 16

Challenge level

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

problem

Contact

Age

14 to 16

Challenge level

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

problem

Out of the window

Age

14 to 16

Challenge level

Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

problem

Star gazing

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.

problem

Mixing more paints

Age

14 to 16

Challenge level

Can you find an efficent way to mix paints in any ratio?

problem

Spotting the loophole

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?

problem

What's that graph?

Age

14 to 18

Challenge level

Can you work out which processes are represented by the graphs?

problem

Concrete calculation

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?

problem

Trisected triangle

Age

14 to 16

Challenge level

Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

article

Proofs with pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

problem

Corridors

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.

problem

Oldest and youngest

Age

14 to 16

Challenge 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?

problem

Picture story

Age

14 to 16

Challenge level

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

problem

Bike shop

Age

14 to 16

Challenge level

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

Five coloured cubes forming the edges of a pentagon.

problem

Penta colour

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?

problem

Areas of parallelograms

Age

14 to 16

Challenge level

Can you find the area of a parallelogram defined by two vectors?

problem

Making tracks

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?

problem

Filling the gaps

Age

14 to 16

Challenge level

Which numbers can we write as a sum of square numbers?

problem

Immersion

Age

14 to 16

Challenge level

Various solids are lowered into a beaker of water. How does the water level rise in each case?

problem

Perception versus reality

Age

14 to 18

Challenge level

Infographics are a powerful way of communicating statistical information. Can you come up with your own?

article

When the angles of a triangle don't add up to 180 degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

problem

Hexy-metry

Age

14 to 16

Challenge level

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

problem

Travelator

Age

14 to 16

Challenge level

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

problem

Proximity

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.

problem

Bus stop

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?

problem

Facial sums

Age

14 to 16

Challenge level

Can you make the numbers around each face of this solid add up to the same total?

problem

Escalator

Age

14 to 16

Challenge level

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners

Visualising - Upper Secondary