List

Visualising - Upper Secondary

Pumpkin Patch
game

Pumpkin Patch

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.
Seega
game

Seega

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.
Alquerque
game

Alquerque

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'.
Introducing NRICH TWILGO
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?
Air Nets
problem

Air Nets

Age
7 to 18
Challenge level
filled star empty star empty star
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Like a Circle in a Spiral
problem

Like a Circle in a Spiral

Age
7 to 16
Challenge level
filled star empty star empty star
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?
Fruity Totals
problem
Favourite

Fruity Totals

Age
7 to 16
Challenge level
filled star empty star empty star

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

Clocking off
problem

Clocking off

Age
7 to 16
Challenge level
filled star empty star empty star
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?
Prime Magic
problem

Prime Magic

Age
7 to 16
Challenge level
filled star filled star empty star
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?
Cubic Conundrum
problem

Cubic Conundrum

Age
7 to 16
Challenge level
filled star filled star filled star
Which of the following cubes can be made from these nets?
Sprouts
game

Sprouts

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
What's it worth?
problem
Favourite

What's it worth?

Age
11 to 16
Challenge level
filled star empty star empty star

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

Nine Colours
problem
Favourite

Nine Colours

Age
11 to 16
Challenge level
filled star filled star filled star
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?
Who's who?
problem

Who's who?

Age
11 to 16
Challenge level
filled star filled star empty star

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

Instant Insanity
problem

Instant Insanity

Age
11 to 18
Challenge level
filled star filled star filled star

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.

Searching for mean(ing)
problem
Favourite

Searching for mean(ing)

Age
11 to 16
Challenge level
filled star filled star empty star
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?
LOGO Challenge - Triangles-Squares-Stars
problem

LOGO Challenge - Triangles-Squares-Stars

Age
11 to 16
Challenge level
filled star filled star empty star
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.
Hamiltonian Cube
problem
Favourite

Hamiltonian Cube

Age
11 to 16
Challenge level
filled star filled star empty star
Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.
Cuboid challenge
problem
Favourite

Cuboid Challenge

Age
11 to 16
Challenge level
filled star filled star empty star

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

Take Three From Five
problem
Favourite

Take Three From Five

Age
11 to 16
Challenge level
filled star filled star empty star

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?

Placeholder: several colourful numbers
problem

Triangles in the middle

Age
11 to 18
Challenge level
filled star empty star empty star
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
11x11 square
problem

11x11 square

Age
11 to 16
Challenge level
filled star filled star empty star
Here's a neat trick you can do with an 11 by 11 square...
What is the question?
problem

What is the question?

Age
11 to 16
Challenge level
filled star empty star empty star
These pictures and answers leave the viewer with the problem "What is the Question". Can you give the question and how the answer follows?
Pythagoras Proofs
problem
Favourite

Pythagoras Proofs

Age
11 to 16
Challenge level
filled star filled star empty star

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

Parallelogram It
problem
Favourite

Parallelogram It

Age
11 to 16
Challenge level
filled star empty star empty star

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.

Charlie's delightful machine
problem
Favourite

Charlie's delightful machine

Age
11 to 16
Challenge level
filled star empty star empty star

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

Charting success
problem

Charting success

Age
11 to 16
Challenge level
filled star empty star empty star
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
Rhombus It
problem
Favourite

Rhombus It

Age
11 to 16
Challenge level
filled star filled star empty star

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.

Sliding Puzzle
game

Sliding Puzzle

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.
Charting more success
problem

Charting more success

Age
11 to 16
Challenge level
filled star empty star empty star
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
Multiple Surprises
problem
Favourite

Multiple Surprises

Age
11 to 16
Challenge level
filled star empty star empty star
Sequences of multiples keep cropping up...
Ding Dong Bell
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.
Marbles in a box
problem
Favourite

Marbles in a box

Age
11 to 16
Challenge level
filled star filled star empty star
How many winning lines can you make in a three-dimensional version of noughts and crosses?
The Bridges of Konigsberg
problem

The Bridges of Konigsberg

Age
11 to 18
Challenge level
filled star empty star empty star
Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.
Shaping the universe I - planet Earth
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.
Semi-regular Tessellations
problem
Favourite

Semi-regular Tessellations

Age
11 to 16
Challenge level
filled star empty star empty star

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

Funny Factorisation
problem
Favourite

Funny Factorisation

Age
11 to 16
Challenge level
filled star filled star empty star
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
Yih or Luk tsut k'i or Three Men's Morris
game

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

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.
Around and Back
problem

Around and Back

Age
14 to 16
Challenge level
filled star filled star filled star
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.
Packing 3D shapes
problem

Packing 3D shapes

Age
14 to 16
Challenge level
filled star filled star filled star
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
Tennis Training
problem

Tennis Training

Age
14 to 16
Challenge level
filled star filled star empty star
After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?
Painted Purple
problem

Painted Purple

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Out of the Window
problem

Out of the Window

Age
14 to 16
Challenge level
filled star empty star empty star
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.
Which is bigger?
problem
Favourite

Which is bigger?

Age
14 to 16
Challenge level
filled star filled star empty star
Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?
Three cubes
problem
Favourite

Three cubes

Age
14 to 16
Challenge level
filled star filled star empty star
Can you work out the dimensions of the three cubes?
Cubic Covering
problem

Cubic Covering

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Eulerian
problem

Eulerian

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Jam
game

Jam

A game for 2 players
Partly Painted Cube
problem
Favourite

Partly Painted Cube

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Bike Shop
problem

Bike Shop

Age
14 to 16
Challenge level
filled star filled star empty star
If I walk to the bike shop, but then cycle back, what is my average speed?
Pyramidal n-gon
problem

Pyramidal n-gon

Age
14 to 16
Challenge level
filled star filled star empty star
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?
AMGM
problem

AMGM

Age
14 to 16
Challenge level
filled star filled star filled star
Can you use the diagram to prove the AM-GM inequality?
Can you traverse it?
problem

Can you traverse it?

Age
14 to 18
Challenge level
filled star empty star empty star

How can you decide if a graph is traversable?

Jam
game

Jam

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

Just rolling round
problem

Just rolling round

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Bus Stop
problem

Bus Stop

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Placeholder: several colourful numbers
problem

Bent out of Shape

Age
14 to 18
Challenge level
filled star filled star empty star
An introduction to bond angle geometry.
Dating made Easier
problem
Favourite

Dating made Easier

Age
14 to 16
Challenge level
filled star filled star filled star
If a sum invested gains 10% each year how long before it has doubled its value?
In or Out?
problem

In or Out?

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Gnomon dimensions
problem

Gnomon dimensions

Age
14 to 16
Challenge level
filled star filled star empty star
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
A question of scale
problem
Favourite

A question of scale

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Vanishing point
problem

Vanishing point

Age
14 to 18
Challenge level
filled star filled star empty star
How can visual patterns be used to prove sums of series?
Wari
game

Wari

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?
Terminology
problem
Favourite

Terminology

Age
14 to 16
Challenge level
filled star filled star empty star
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Coke machine
problem

Coke machine

Age
14 to 16
Challenge level
filled star empty star empty star
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...
Inside Out
problem

Inside Out

Age
14 to 16
Challenge level
filled star filled star empty star

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?

Vector journeys
problem
Favourite

Vector journeys

Age
14 to 18
Challenge level
filled star empty star empty star
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?
Doesn't add up
problem
Favourite

Doesn't add up

Age
14 to 16
Challenge level
filled star filled star empty star

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

Trisected Triangle
problem

Trisected Triangle

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
Just Opposite
problem

Just Opposite

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Sliced
problem

Sliced

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Factorising with Multilink
problem
Favourite

Factorising with Multilink

Age
14 to 16
Challenge level
filled star empty star empty star
Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?
Fermat's Poser
problem

Fermat's Poser

Age
14 to 16
Challenge level
filled star filled star filled star
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
Rectangle Rearrangement
problem

Rectangle Rearrangement

Age
14 to 16
Challenge level
filled star empty star empty star
A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?
Quadratic Patterns
problem
Favourite

Quadratic Patterns

Age
14 to 16
Challenge level
filled star empty star empty star

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

Iff
problem
Favourite

Iff

Age
14 to 18
Challenge level
filled star filled star empty star
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
At right angles
problem
Favourite

At right angles

Age
14 to 16
Challenge level
filled star filled star empty star
Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?
Back fitter
problem
Favourite

Back fitter

Age
14 to 18
Challenge level
filled star filled star empty star

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?

Pick's Theorem
problem
Favourite

Pick's Theorem

Age
14 to 16
Challenge level
filled star filled star empty star
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.
Building Tetrahedra
problem

Building Tetrahedra

Age
14 to 16
Challenge level
filled star filled star empty star
Can you make a tetrahedron whose faces all have the same perimeter?
Circuit training
problem

Circuit training

Age
14 to 16
Challenge level
filled star empty star empty star
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?
What's that graph?
problem
Favourite

What's that graph?

Age
14 to 18
Challenge level
filled star filled star empty star

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

A Problem of time
problem

A Problem of time

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Pythagoras Perimeters
problem
Favourite

Pythagoras Perimeters

Age
14 to 16
Challenge level
filled star filled star empty star
If you know the perimeter of a right angled triangle, what can you say about the area?
Tetra Square
problem

Tetra Square

Age
14 to 18
Challenge level
filled star filled star empty star
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.
Building Gnomons
problem

Building Gnomons

Age
14 to 16
Challenge level
filled star empty star empty star
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Surprising Transformations
problem
Favourite

Surprising Transformations

Age
14 to 16
Challenge level
filled star filled star empty star

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?

Corridors
problem

Corridors

Age
14 to 16
Challenge level
filled star filled star empty star
A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
Something in Common
problem

Something in Common

Age
14 to 16
Challenge level
filled star filled star filled star
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.
Filling the gaps
problem
Favourite

Filling the gaps

Age
14 to 16
Challenge level
filled star filled star empty star
Which numbers can we write as a sum of square numbers?
Hypotenuse Lattice points
problem

Hypotenuse Lattice points

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Concrete calculation
problem

Concrete calculation

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Always Perfect
problem
Favourite

Always Perfect

Age
14 to 18
Challenge level
filled star filled star empty star
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
3D Treasure Hunt
problem

3D Treasure Hunt

Age
14 to 18
Challenge level
filled star filled star empty star
Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?
Star Gazing
problem

Star Gazing

Age
14 to 16
Challenge level
filled star empty star empty star
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
Twelve Cubed
problem

Twelve Cubed

Age
14 to 16
Challenge level
filled star empty star empty star
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?