The points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Four rods of equal length are hinged at their endpoints to form a rhombus. The diagonals meet at X. One edge is fixed, the opposite edge is allowed to move in the plane. Describe the locus of. . . .

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

Points D, E and F are on the the sides of triangle ABC. Circumcircles are drawn to the triangles ADE, BEF and CFD respectively. What do you notice about these three circumcircles?

Charlie has created a mapping. Can you figure out what it does? What questions does it prompt you to ask?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Alison has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

Drawing a triangle is not always as easy as you might think!

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

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. . . .

Explore the relationship between quadratic functions and their graphs.

Show that all pentagonal numbers are one third of a triangular number.

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Can you find a rule which connects consecutive triangular numbers?

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Find out about Magic Squares in this article written for students. Why are they magic?!

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. . . .

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.

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

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

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Can you find a rule which relates triangular numbers to square numbers?

A introduction to how patterns can be deceiving, and what is and is not a proof.

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

Explore the relationship between simple linear functions and their graphs.