Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

Are these statements always true, sometimes true or never true?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

Is it possible to draw a 5-pointed star without taking your pencil off the paper? Is it possible to draw a 6-pointed star in the same way without taking your pen off?

Here are two kinds of spirals for you to explore. What do you notice?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

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

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

An investigation that gives you the opportunity to make and justify predictions.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

This challenge combines addition, multiplication, perseverance and even proof.

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

This task combines spatial awareness with addition and multiplication.

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

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

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

Use the information about the ducks on a particular farm to find out which of the statements about them must be true.

Investigate the totals you get when adding numbers on the diagonal of this pattern in threes.

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

In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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.

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. . . .

Are these statements always true, sometimes true or never true?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.