This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

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.

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

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

This problem looks at how one example of your choice can show something about the general structure of multiplication.

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

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?

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.

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

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

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

Proof does have a place in Primary mathematics classrooms, we just need to be clear about what we mean by proof at this level.

This investigates one particular property of number by looking closely at an example of adding two odd numbers together.