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.

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

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?

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.

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.

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 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?

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. . . .

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

Can you prove that twice the sum of two squares always gives the sum of two squares?

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

Relate these algebraic expressions to geometrical diagrams.

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

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

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 the five distinct digits N, R, I, C and H in the following nomogram

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

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

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

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

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.

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

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?

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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 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.

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

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

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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...

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. . . .

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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

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

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

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.

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

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Can you explain what is going on in these puzzling number tricks?

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Can you see how to build a harmonic triangle? Can you work out the next two rows?