For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
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 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
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?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Small circles nestle under touching parent circles when they sit on
the axis at neighbouring points in a Farey sequence.
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 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. . . .
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
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
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?
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.
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 game for 2 players
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?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
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. . . .
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
How efficiently can you pack together disks?
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?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
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.
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.
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?
What can you see? What do you notice? What questions can you ask?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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?
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?
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?
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
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 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. . . .
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
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 ?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Can you discover whether this is a fair game?
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.
See if you can anticipate successive 'generations' of the two
animals shown here.
Can you describe this route to infinity? Where will the arrows take you next?
Show that all pentagonal numbers are one third of a triangular number.
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 connects consecutive triangular numbers?