Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

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 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 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.

Five equations... five unknowns... can you solve the system?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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

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 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.

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

An algebra task which depends on members of the group noticing the needs of others and responding.

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

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

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?

Relate these algebraic expressions to geometrical diagrams.

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

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

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

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. . . .

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

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 a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

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

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which connects consecutive triangular numbers?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

Can you find a rule which relates triangular numbers to square numbers?

How good are you at finding the formula for a number pattern ?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Can you find the lap times of the two cyclists travelling at constant speeds?

Can you hit the target functions using a set of input functions and a little calculus and algebra?

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.