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.

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

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.

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

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

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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.

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

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?

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

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

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

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. . . .

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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...

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.

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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?

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

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

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

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.

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

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.

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. . . .

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

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

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

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

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.

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. . . .

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.

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.

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

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

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

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

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