What 3D shapes occur in nature. How efficiently can you pack these shapes together?
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. . . .
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.
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
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 ?
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
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?
An introduction to bond angle geometry.
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.
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. . . .
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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.
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.
Can you find a rule which relates triangular numbers to square numbers?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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?
Can you find a rule which connects consecutive triangular numbers?
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
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?
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?
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?
To avoid losing think of another very well known game where the
patterns of play are similar.
A game for 2 players
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?
See if you can anticipate successive 'generations' of the two
animals shown here.
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
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?
What can you see? What do you notice? What questions can you ask?
Show that all pentagonal numbers are one third of a triangular number.
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. . . .
A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
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?
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. . . .
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .
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.
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.
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 visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
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?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
Two intersecting circles have a common chord AB. The point C moves
on the circumference of the circle C1. The straight lines CA and CB
meet the circle C2 at E and F respectively. As the point C. . . .
This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.