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?

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.

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

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?

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?

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.

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?

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.

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

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.

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. . . .

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Prove Pythagoras' Theorem using enlargements and scale factors.

Which of these triangular jigsaws are impossible to finish?

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?

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

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

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.

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?

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

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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.

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?

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

Do you have enough information to work out the area of the shaded quadrilateral?

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!

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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.

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

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

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

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

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.

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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. . . .

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?