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An account of multiplication of vectors, both scalar products and vector products.

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

Yatir from Israel describes his method for summing a series of triangle numbers.

The harmonic triangle is built from fractions with unit numerators using a rule very similar to Pascal's triangle.

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.

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.

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.

What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

Some statements which can be proved using induction, and some example proofs.

Selection of STEP Statistics questions with hints for students to undertake as part of the Statistics STEP Prep Module.

Some simple ideas about graph theory with a discussion of a proof of Euler's formula relating the numbers of vertces, edges and faces of a graph.

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

Class 2YP from Madras College was inspired by the problem in NRICH to work out in how many ways the number 1999 could be expressed as the sum of 3 odd numbers, and this is their solution.

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. . . .

Some of our experiences of discovering and using triangle numbers in a range of contexts.

In this article, Alan Parr shares his experiences of the motivating effect sport can have on the learning of mathematics.

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.

Infinity is not a number, and trying to treat it as one tends to be a pretty bad idea. At best you're likely to come away with a headache, at worse the firm belief that 1 = 0. This article discusses. . . .

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

Is it possible to make an irregular polyhedron using only polygons of, say, six, seven and eight sides? The answer (rather surprisingly) is 'no', but how do we prove a statement like this?

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.

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.

This article gives an introduction to mathematical induction, a powerful method of mathematical proof.

This article examines how probability can often be viewed simply as counting the number of ways certain events can occur, before covering some summarising examples.

Need some help getting started with solving and thinking about rich tasks? Read on for some friendly advice.

In STEP and other advanced mathematics examinations, two main methods are used to solve first order differential equations: separation of variables and the method of integrating factors. This article. . . .

How much are you likely to win from a raffle? How many loops will you make with some strings? Here, two guided examples can be found for you to work through.

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

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

A short introduction to complex numbers written primarily for students aged 14 to 19.

How high can a high jumper jump? How can a high jumper jump higher without jumping higher? Read on...

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

Step back and reflect! This article reviews techniques such as substitution and change of coordinates which enable us to exploit underlying structures to crack problems.

Having a good grasp of trigonometry is important for solving advanced mathematics examinations questions.

A introduction to how patterns can be deceiving, and what is and is not a proof.

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.

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

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

When in 1821 Charles Babbage invented the `Difference Engine' it was intended to take over the work of making mathematical tables by the techniques described in this article.

Sanjay Joshi, age 17, The Perse Boys School, Cambridge followed up the Madrass College class 2YP article with more thoughts on the problem of the number of ways of expressing an integer as the sum. . . .

Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.

In this article, Rachel Melrose describes what happens when she mixed mathematics with art.

Ranging from kindergarten mathematics to the fringe of research this informal article paints the big picture of number in a non technical way suitable for primary teachers and older students.

How do decisions about scoring affect who wins a combined event such as the decathlon?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.