Have you got the Mach knack? Discover the mathematics behind
exceeding the sound barrier.
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
How efficiently can various flat shapes be fitted together?
The net of a cube is to be cut from a sheet of card 100 cm square.
What is the maximum volume cube that can be made from a single
piece of card?
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?
A circular plate rolls inside a rectangular tray making five
circuits and rotating about its centre seven times. Find the
dimensions of the tray.
Can you work out the dimensions of the three cubes?
An introduction to bond angle geometry.
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 ?
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
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?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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?
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. . . .
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?
A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?
What can you see? What do you notice? What questions can you ask?
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?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
How efficiently can you pack together disks?
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?
Can you make a tetrahedron whose faces all have the same perimeter?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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.
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!
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
A triangle PQR, right angled at P, slides on a horizontal floor
with Q and R in contact with perpendicular walls. What is the locus
In this problem we see how many pieces we can cut a cube of cheese
into using a limited number of slices. How many pieces will you be
able to make?
Takes you through the systematic way in which you can begin to
solve a mixed up Cubic Net. How close will you come to a solution?
Some diagrammatic 'proofs' of algebraic identities and
A bicycle passes along a path and leaves some tracks. Is it
possible to say which track was made by the front wheel and which
by the back wheel?
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 visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
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.
Simple additions can lead to intriguing results...
See if you can anticipate successive 'generations' of the two
animals shown here.
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
Square It game for an adult and child. Can you come up with a way of always winning this game?
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. . . .
In how many different ways can I colour the five edges of a
pentagon red, blue and green so that no two adjacent edges are the
Can you find a rule which connects consecutive triangular numbers?
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. . . .
Can you find a rule which relates triangular numbers to square numbers?
Show that all pentagonal numbers are one third of a triangular number.
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.