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 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.

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.

Relate these algebraic expressions to geometrical diagrams.

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.

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.

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?

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.

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

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. . . .

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

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.

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. . . .

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.

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}.

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

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

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.

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

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

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

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

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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.

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?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

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. . . .

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.

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

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

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?

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

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. . . .

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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

Can you rearrange the cards to make a series of correct mathematical statements?

Explore a number pattern which has the same symmetries in different bases.