Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
How efficiently can various flat shapes be fitted together?
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.
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?
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. . . .
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
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.
How efficiently can you pack together disks?
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
Have you got the Mach knack? Discover the mathematics behind
exceeding the sound barrier.
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. . . .
To avoid losing think of another very well known game where the
patterns of play are similar.
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
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. . . .
Can you describe this route to infinity? Where will the arrows take you next?
For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Can you find a rule which relates triangular numbers to square numbers?
See if you can anticipate successive 'generations' of the two
animals shown here.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
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.
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
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
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
A game for 2 players
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?
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
Show that all pentagonal numbers are one third of a triangular number.
Simple additions can lead to intriguing results...
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.
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. . . .
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
Can you find a rule which connects consecutive triangular numbers?
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?
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.
Mark a point P inside a closed curve. Is it always possible to find
two points that lie on the curve, such that P is the mid point of
the line joining these two points?
A circular plate rolls inside a rectangular tray making five
circuits and rotating about its centre seven times. Find the
dimensions of the tray.
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.
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. . . .
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
Square It game for an adult and child. Can you come up with a way of always winning this game?
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
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. . . .
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
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.