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

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.

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.

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?

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.

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. . . .

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

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

Relate these algebraic expressions to geometrical diagrams.

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

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.

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

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

An inequality involving integrals of squares of functions.

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?

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}.

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.

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

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. . . .

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

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

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

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

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.

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

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

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

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

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.

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.

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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. . . .

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 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.

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

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.

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

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

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.

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

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...

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

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.

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 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.