Visualising - Upper Secondary

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'.

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?

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

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

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

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

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

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

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

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

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

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

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

11x11 square

Age

11 to 16

Challenge level

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

game

The triangle game

Age

11 to 16

Challenge level

Can you discover whether this is a fair 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

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

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

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

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?

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

Cuboid challenge

Age

11 to 16

Challenge level

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

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

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

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?

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

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

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

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

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

Who's who?

Age

11 to 16

Challenge level

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

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

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.

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.

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

Can you traverse it?

Age

14 to 18

Challenge level

How can you decide if a graph is traversable?

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

Cubic covering

Age

14 to 16

Challenge level

A blue cube has blue cubes glued on all of its faces. Yellow cubes are then glued onto all the visible blue facces. How many yellow cubes are needed?

problem

What's that graph?

Age

14 to 18

Challenge level

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

problem

Quadratic patterns

Age

14 to 16

Challenge level

Surprising numerical patterns can be explained using algebra and diagrams...

game

Jam

Age

14 to 16

Challenge level

A game for 2 players

problem

The root of the problem

Age

14 to 18

Challenge level

Find the sum of this series of surds.

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

Vanishing point

Age

14 to 18

Challenge level

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

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

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

Filling the gaps

Age

14 to 16

Challenge level

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

problem

Around and back

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.

problem

Efficient packing

Age

14 to 16

Challenge level

How efficiently can you pack together disks?

problem

Pythagoras perimeters

Age

14 to 16

Challenge level

If you know the perimeter of a right angled triangle, what can you say about the area?

game

Jam

Age

14 to 16

Challenge level

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

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

Bike shop

Age

14 to 16

Challenge level

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

problem

Triangle midpoints

Age

14 to 16

Challenge level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

problem

Nicely similar

Age

14 to 16

Challenge level

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

problem

Packing 3D shapes

Age

14 to 16

Challenge level

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

game

Wari

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?

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

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

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

Curve hunter

Age

14 to 18

Challenge level

This problem challenges you to sketch curves with different properties.

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

Dating made easier

Age

14 to 16

Challenge level

If a sum invested gains 10% each year how long before it has doubled its value?

problem

AMGM

Age

14 to 16

Challenge level

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

problem

Quadratic matching

Age

14 to 16

Challenge level

Can you match each graph to one of the statements?

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

Building tetrahedra

Age

14 to 16

Challenge level

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

problem

Inside out

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?

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

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

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

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

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

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

Sliced

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?

problem

Centre square

Age

14 to 16

Challenge level

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

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

Rectangle rearrangement

Age

14 to 16

Challenge 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?

problem

Summing geometric progressions

Age

14 to 18

Challenge level

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

problem

Square number surprises

Age

14 to 16

Challenge level

There are unexpected discoveries to be made about square numbers...

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

Circuit training

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?

problem

Folding in half

Age

14 to 16

Challenge level

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

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

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

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

Puzzling place value

Age

14 to 16

Challenge level

Can you explain what is going on in these puzzling number tricks?

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

Something in common

Age

14 to 16

Challenge level

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.