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.

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, 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?

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.

Relate these algebraic expressions to geometrical diagrams.

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

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

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

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. . . .

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?

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 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?

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

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

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?

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. . . .

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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

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

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...

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

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.

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

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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.

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

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. . . .

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

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

A introduction to how patterns can be deceiving, and what is and is not a proof.

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. . . .

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

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

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. . . .

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?

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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.