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

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

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.

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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.

How many noughts are at the end of these giant numbers?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Do you have enough information to work out the area of the shaded quadrilateral?

Can the pdfs and cdfs of an exponential distribution intersect?

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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...

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

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

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.

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

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.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

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

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

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.

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

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. . . .

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

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

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

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

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.

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Prove Pythagoras' Theorem using enlargements and scale factors.

The sums of the squares of three related numbers is also a perfect square - can you explain why?