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

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

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

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Keep constructing triangles in the incircle of the previous triangle. What happens?

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

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.

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

Prove Pythagoras' Theorem using enlargements and scale factors.

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Kyle and his teacher disagree about his test score - who is right?

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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

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.