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

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

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

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.

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

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

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.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

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.

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

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

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

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

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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. . . .

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...

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

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}.

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.

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

Relate these algebraic expressions to geometrical diagrams.

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.

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

When is it impossible to make number sandwiches?

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

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.

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.

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

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.

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.

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?

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

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

An introduction to some beautiful results of Number Theory

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

Have a go at being mathematically negative, by negating these statements.

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.

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.