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?

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

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

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

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

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?

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

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

This task combines spatial awareness with addition and multiplication.

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

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

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

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?

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

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?

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

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?

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

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

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.

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

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. . . .

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

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

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

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

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 ...

Explore the relationship between simple linear functions and their graphs.

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.

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

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