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.
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.
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'.
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
Favourite
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
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
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
Favourite
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
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
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
11x11 square
Age
11 to 16
Challenge level
Here's a neat trick you can do with an 11 by 11 square...
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
Favourite
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
Favourite
Pythagoras proofs
Age
11 to 16
Challenge level
Can you make sense of these three proofs of Pythagoras' Theorem?
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.
problem
Favourite
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?
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
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
Favourite
Multiple surprises
Age
11 to 16
Challenge level
Sequences of multiples keep cropping up...
game
Sprouts
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
problem
Favourite
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?
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 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
Favourite
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
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
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
Favourite
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
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
Favourite
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.
Find the length along the shortest path passing through certain points on the cube.
problem
Favourite
Cuboid challenge
Age
11 to 16
Challenge level
What's the largest volume of box you can make from a square of paper?
problem
Favourite
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
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?
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.
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
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
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?
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
problem
Favourite
One and three
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?
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
Can you traverse it?
Age
14 to 18
Challenge level
How can you decide if a graph is traversable?
problem
Overlap
Age
14 to 16
Challenge level
A red square and a blue square overlap. Is the area of the overlap always the same?
problem
Favourite
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
Favourite
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
Favourite
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
Triangles within squares
Age
14 to 16
Challenge level
Can you find a rule which relates triangular numbers to square numbers?
problem
Favourite
Quadratic patterns
Age
14 to 16
Challenge level
Surprising numerical patterns can be explained using algebra and diagrams...
problem
Favourite
Plus minus
Age
14 to 16
Challenge level
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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
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
Favourite
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
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
Triangles within pentagons
Age
14 to 16
Challenge level
Show that all pentagonal numbers are one third of a triangular number.
problem
Favourite
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?
problem
Favourite
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
Favourite
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?
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
Favourite
What's that graph?
Age
14 to 18
Challenge level
Can you work out which processes are represented by the graphs?
problem
Natural sum
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.
problem
Favourite
Mixing more paints
Age
14 to 16
Challenge level
Can you find an efficent way to mix paints in any ratio?
problem
Favourite
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
Favourite
Attractive tablecloths
Age
14 to 16
Challenge level
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
problem
Favourite
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
Favourite
Filling the gaps
Age
14 to 16
Challenge level
Which numbers can we write as a sum of square numbers?
problem
Quadratic matching
Age
14 to 16
Challenge level
Can you match each graph to one of the statements?
problem
Favourite
The root of the problem
Age
14 to 18
Challenge level
Find the sum of this series of surds.
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
Platonic planet
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?
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
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
Packing 3D shapes
Age
14 to 16
Challenge level
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
problem
Favourite
Vector walk
Age
14 to 18
Challenge level
Starting with two basic vector steps, which destinations can you reach on a vector walk?
problem
Curve hunter
Age
14 to 18
Challenge level
This problem challenges you to sketch curves with different properties.
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
Favourite
Three cubes
Age
14 to 16
Challenge level
Can you work out the dimensions of the three cubes?
problem
Favourite
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
Favourite
Square number surprises
Age
14 to 16
Challenge level
There are unexpected discoveries to be made about square numbers...
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
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
Favourite
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
Favourite
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
Favourite
Puzzling place value
Age
14 to 16
Challenge level
Can you explain what is going on in these puzzling number tricks?
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?
game
Jam
To avoid losing think of another very well known game where the patterns of play are similar.
problem
Centre square
Age
14 to 16
Challenge level
What does Pythagoras' Theorem tell you about the radius of these circles?
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
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
Favourite
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
Favourite
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
Favourite
Hollow squares
Age
14 to 16
Challenge level
Which armies can be arranged in hollow square fighting formations?
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
Favourite
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?
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?