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 largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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. . . .

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 diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Can you find the areas of the trapezia in this sequence?

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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. . . .

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Keep constructing triangles in the incircle of the previous triangle. What happens?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Prove Pythagoras' Theorem using enlargements and scale factors.

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

Can you make sense of these three proofs of Pythagoras' Theorem?

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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?

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Can you make sense of the three methods to work out the area of the kite in the square?

Can you work through these direct proofs, using our interactive proof sorters?

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

Kyle and his teacher disagree about his test score - who is right?