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
Can you discover whether this is a fair game?
Can you describe this route to infinity? Where will the arrows take you next?
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
This article for teachers discusses examples of problems in which
there is no obvious method but in which children can be encouraged
to think deeply about the context and extend their ability to. . . .
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
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.
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.
Square It game for an adult and child. Can you come up with a way of always winning this game?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
Can you maximise the area available to a grazing goat?
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.
Draw a pentagon with all the diagonals. This is called a pentagram.
How many diagonals are there? How many diagonals are there in a
hexagram, heptagram, ... Does any pattern occur when looking at. . . .
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
An activity centred around observations of dots and how we visualise number arrangement patterns.
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
An extension of noughts and crosses in which the grid is enlarged
and the length of the winning line can to altered to 3, 4 or 5.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start.
How many Hamiltonian circuits can you find in these graphs?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Which of these dice are right-handed and which are left-handed?
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
Can you find a way of representing these arrangements of balls?
This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
Can you fit the tangram pieces into the outlines of the chairs?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
Can you fit the tangram pieces into the outlines of these clocks?
Can you fit the tangram pieces into the outline of the child walking home from school?
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?