Make some loops out of regular hexagons. What rules can you discover?

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

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

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?

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

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. . . .

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?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

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

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

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

Can you find a rule which connects consecutive triangular numbers?

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

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?

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?

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

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.

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

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

What's the greatest number of sides a polygon on a dotty grid could have?

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?

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.