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?
For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
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 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. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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?
Small circles nestle under touching parent circles when they sit on
the axis at neighbouring points in a Farey sequence.
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
Can you use the diagram to prove the AM-GM inequality?
To avoid losing think of another very well known game where the
patterns of play are similar.
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. . . .
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?
A game for 2 players
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?
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
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?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
See if you can anticipate successive 'generations' of the two
animals shown here.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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?
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
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
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.
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?
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 happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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?
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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
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.
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?
A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
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 connects consecutive triangular numbers?
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.
Simple additions can lead to intriguing results...
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 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?
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
Show that all pentagonal numbers are one third of a triangular number.
Can you describe this route to infinity? Where will the arrows take you next?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
Use the diagram to investigate the classical Pythagorean means.