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. . . .

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 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.

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.

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

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 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.

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

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

Relate these algebraic expressions to geometrical diagrams.

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

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. . . .

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

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

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.

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?

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

I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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.

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.

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.

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...

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

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

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

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.

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.

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

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. . . .

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...

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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.

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.

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

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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 orf two or more cubes.

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

By proving these particular identities, prove the existence of general cases.

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

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

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

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?