Search by Topic

Find articles and problems on the site

Find articles and problems on the site by choosing curriculum topics.
Each problem page displayed gives a list of similar problems and a list of topics to help you find more.
Alternatively see our previous monthly themes.

1. Choose a broad topic

• Using, Applying and Reasoning about Mathematics

2. Choose a detailed topic

• Mathematical reasoning & proof

3. Choose a resource

There are 242 results

Tower of Hanoi

Stage: 4 Challenge Level:

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Breaking the Equation ' Empirical Argument = Proof '

Stage: 2, 3, 4 and 5

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.

The Clue Is in the Question

Stage: 5 Challenge Level:

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

Cyclic Quadrilaterals

Stage: 3 Challenge Level:

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

Sticky Numbers

Stage: 3 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Pythagoras Proofs

Stage: 4 Challenge Level:

Can you make sense of these three proofs of Pythagoras' Theorem?

Take a Square II

Stage: 4 Challenge Level:

What fractions can you divide the diagonal of a square into by simple folding?

Power Mad!

Stage: 3 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Elevenses

Stage: 3 Challenge Level:

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Mind Your Ps and Qs

Stage: 5 Short Challenge Level:

Sort these mathematical propositions into a series of 8 correct statements.

NRICH Starter Problem Selection

Stage: Challenge Level:

On this page we give a selection of good starter activities for those new to NRICH

Notty Logic

Stage: 5 Challenge Level:

Have a go at being mathematically negative, by negating these statements.

Direct Logic

Stage: 5 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?

Contrary Logic

Stage: 5 Challenge Level:

Can you invert the logic to prove these statements?

Iffy Logic

Stage: 4 Short Challenge Level:

Can you rearrange the cards to make a series of correct mathematical statements?

Air Nets

Stage: 2, 3, 4 and 5 Challenge Level:

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

Truth Tables and Electronic Circuits

Stage: 2, 3 and 4

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

Logic, Truth Tables and Switching Circuits Challenge

Stage: 2, 3, 4 and 5

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to. . . .

Proof: A Brief Historical Survey

Stage: 4 and 5

If you think that mathematical proof is really clearcut and universal then you should read this article.

What Do You Need?

Stage: 2 Challenge Level:

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?

Cows and Sheep

Stage: 2 Challenge Level:

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

A Bag of Marbles

Stage: 1 Challenge Level:

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

Road Maker 2

Stage: 5 Short Challenge Level:

Can you work out where the blue-and-red brick roads end?

Impossible Triangles?

Stage: 5 Challenge Level:

Which of these triangular jigsaws are impossible to finish?

Road Maker

Stage: 4 and 5 Challenge Level:

Which of these roads will satisfy a Munchkin builder?

9 Weights

Stage: 3 Challenge Level:

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

The Great Weights Puzzle

Stage: 4 Challenge Level:

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Symmetric Tangles

Stage: 4

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 Challenge Level:

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.

Proof of Pick's Theorem

Stage: 5 Challenge Level:

Follow the hints and prove Pick's Theorem.

Integral Inequality

Stage: 5 Challenge Level:

An inequality involving integrals of squares of functions.

Tessellating Hexagons

Stage: 3 Challenge Level:

Is it true that any convex hexagon will tessellate if it has a pair of opposite sides that are equal, and three adjacent angles that add up to 360 degrees?

Concrete Wheel

Stage: 3 Challenge Level:

A huge wheel is rolling past your window. What do you see?

Introductory Number Theory

Stage: 4 and 5

An introduction to some beautiful results of Number Theory

Primary Proof?

Stage: 1

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

Multiplication Square

Stage: 3 Challenge Level:

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Plus or Minus

Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers F{n+1}F{n-1}.

The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Golden Eggs

Stage: 5 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Binary Sequences

Stage: 5 Challenge Level:

Show that the infinite set of finite (or terminating) binary sequences can be written as an ordered list whereas the infinite set of all infinite binary sequences cannot.

Shuffle Shriek

Stage: 3 Challenge Level:

Can you find all the 4-ball shuffles?

Logic

Stage: 2 and 3

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

Paradoxes

Stage: 2

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.

Salinon

Stage: 4 Challenge Level:

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

And So on - and on -and On

Stage: 5 Challenge Level:

Can you find the value of this function involving algebraic fractions for x=2000?

Some Circuits in Graph or Network Theory

Stage: 4 and 5

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

Sprouts

Stage: 2, 3, 4 and 5

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. . . .

Can it Be

Stage: 5 Challenge Level:

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Thousand Words

Stage: 5 Challenge Level:

Here the diagram says it all. Can you find the diagram?

Cube Net

Stage: 5 Challenge Level:

How many tours visit each vertex of a cube once and only once? How many return to the starting point?