Visualising - Upper Secondary

interactivity

Introducing NRICH TWILGO

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?

problem

Baravelle

Age

7 to 16

Challenge level

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

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

Cubic Conundrum

Age

7 to 16

Challenge level

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

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

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

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?

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

Parallelogram It

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.

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.

problem

Semi-regular Tessellations

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?

problem

Rhombus It

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.

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?

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

11x11 square

Age

11 to 16

Challenge level

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

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?

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

LOGO Challenge - Circles as animals

Age

11 to 16

Challenge level

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

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

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

Cuboid challenge

Age

11 to 16

Challenge level

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

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.

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.

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

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?

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

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

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

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?

problem

Twelve Cubed

Age

14 to 16

Challenge 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?

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

AMGM

Age

14 to 16

Challenge level

Can you use the diagram to prove the AM-GM inequality?

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

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

Just rolling round

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?

problem

3D Treasure Hunt

Age

14 to 18

Challenge level

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

problem

Steel Cables

Age

14 to 16

Challenge level

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

problem

Doesn't add up

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?

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

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

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

Speeding boats

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?

problem

Always Perfect

Age

14 to 18

Challenge level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

problem

Centre Square

Age

14 to 16

Challenge level

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

problem

Coke machine

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...

problem

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?

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

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

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

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

Back fitter

Age

14 to 18

Challenge level

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

problem

Perfectly Square

Age

14 to 16

Challenge level

The sums of the squares of three related numbers is also a perfect square - can you explain why?

problem

Folding in Half

Age

14 to 16

Challenge level

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

problem

Just Opposite

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?

problem

Double Trouble

Age

14 to 16

Challenge level

Simple additions can lead to intriguing results...

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

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

Travelator

Age

14 to 16

Challenge level

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

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

Changing Places

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?

problem

Packing Boxes

Age

14 to 16

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.

problem

Building Tetrahedra

Age

14 to 16

Challenge level

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

problem

Triangles within Triangles

Age

14 to 16

Challenge level

Can you find a rule which connects consecutive triangular numbers?

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

One Out One Under

Age

14 to 16

Challenge level

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

problem

Tennis Training

Age

14 to 16

Challenge level

After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?

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

Surprising Transformations

Age

14 to 16

Challenge level

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

problem

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?

problem

Newspaper Sheets

Age

14 to 16

Challenge level

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

problem

Triangles within Squares

Age

14 to 16

Challenge level

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

problem

Triangles within Pentagons

Age

14 to 16

Challenge level

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

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

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

The perforated cube

Age

14 to 16

Challenge level

A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

problem

Vanishing point

Age

14 to 18

Challenge level

How can visual patterns be used to prove sums of series?

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

Tetrahedra Tester

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?

problem

Speed-time problems at the Olympics

Age

14 to 16

Challenge level

Have you ever wondered what it would be like to race against Usain Bolt?

problem

Mixing More Paints

Age

14 to 16

Challenge level

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

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

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

Negatively Triangular

Age

14 to 16

Challenge level

How many intersections do you expect from four straight lines ? Which three lines enclose a triangle with negative co-ordinates for every point ?

problem

Tilting Triangles

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?

problem

Which is cheaper?

Age

14 to 16

Challenge level

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

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

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

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?

problem

A Tilted Square

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?

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

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

Which is bigger?

Age

14 to 16

Challenge level

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

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?