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. . . .
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
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?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Can you discover whether this is a fair game?
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
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. . . .
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
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 game for 2 players
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 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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
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. . . .
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. . . .
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
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?
Can you describe this route to infinity? Where will the arrows take you next?
To avoid losing think of another very well known game where the
patterns of play are similar.
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?
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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.
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
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.
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?
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.
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?
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?
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. . . .
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
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. . . .
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. . . .
Can you maximise the area available to a grazing goat?
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight