Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Every day at noon a boat leaves Le Havre for New York while another
boat leaves New York for Le Havre. The ocean crossing takes seven
days. How many boats will each boat cross during their journey?
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?
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
A game for 2 players
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
How efficiently can you pack together disks?
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. . . .
A bus route has a total duration of 40 minutes. Every 10 minutes,
two buses set out, one from each end. How many buses will one bus
meet on its way from one end to the other end?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
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. . . .
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
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. . . .
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
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?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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?
Simple additions can lead to intriguing results...
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
Square It game for an adult and child. Can you come up with a way of always winning this game?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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.
At the time of writing the hour and minute hands of my clock are at
right angles. How long will it be before they are at right angles
To avoid losing think of another very well known game where the
patterns of play are similar.
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?
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?
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.
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. . . .
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?
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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.
What is the shape of wrapping paper that you would need to completely wrap this model?
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?
What can you see? What do you notice? What questions can you ask?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
See if you can anticipate successive 'generations' of the two
animals shown here.
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.
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 square.
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
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!
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.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.