How efficiently can various flat shapes be fitted together?
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
Have you got the Mach knack? Discover the mathematics behind
exceeding the sound barrier.
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 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 ?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
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.
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. . . .
How efficiently can you pack together disks?
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.
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.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?
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.
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?
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
A game for 2 players
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?
Small circles nestle under touching parent circles when they sit on
the axis at neighbouring points in a Farey sequence.
Can you make a tetrahedron whose faces all have the same perimeter?
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?
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 game for 2 people. Take turns joining two dots, until your opponent is unable to move.
What can you see? What do you notice? What questions can you ask?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
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?
You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.
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 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?
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
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?
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?
A circular plate rolls inside a rectangular tray making five
circuits and rotating about its centre seven times. Find the
dimensions of the tray.
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?
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.
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Show that all pentagonal numbers are one third of a triangular number.
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
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.
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
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?
Can you find a rule which relates triangular numbers to square numbers?
Can you find a rule which connects consecutive triangular numbers?
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. . . .