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.

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?

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

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. . . .

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?

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

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

Relate these algebraic expressions to geometrical diagrams.

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?

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

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

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

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

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?

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

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?

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 convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

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

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.

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?

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

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

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

There are unexpected discoveries to be made about square numbers...

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?

What is special about the difference between squares of numbers adjacent to multiples of three?

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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

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

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.

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

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.

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