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

Pick a square within a multiplication square and add the numbers on each diagonal. 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?

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?

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?

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.

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?

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

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

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

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}.

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?

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

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. . . .

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

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

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

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

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

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

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

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. . . .

Explore the relationship between quadratic functions and their graphs.

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

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.

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?

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?

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?

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

Can you find a rule which connects consecutive triangular numbers?

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. . . .

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

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

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

Explore the relationship between simple linear functions and their graphs.

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.