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.

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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

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

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Relate these algebraic expressions to geometrical diagrams.

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Can you find the value of this function involving algebraic fractions for x=2000?

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

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Four jewellers share their stock. Can you work out the relative values of their gems?

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

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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.

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. . . .

Can you explain why a sequence of operations always gives you perfect squares?

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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

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

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

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.

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

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.

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.

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

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.

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

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.

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.

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

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