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.

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. . . .

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?

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. . . .

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.

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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?

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. . . .

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?

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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

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?

Prove Pythagoras' Theorem using enlargements and scale factors.

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

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

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.

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.

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 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!

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?

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.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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.

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. . . .

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

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

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.

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

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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?

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

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

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.

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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

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

An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)

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?

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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