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

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

Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .

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

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

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

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

An inequality involving integrals of squares of functions.

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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 record. . . .

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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

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.

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.

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.

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

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

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...

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

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.

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. . . .

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.

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.

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

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.

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

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

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

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

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?

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.

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

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.

Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.

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

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.

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.

An article which gives an account of some properties of magic squares.

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.

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

Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...

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. . . .