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

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

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.

Relate these algebraic expressions to geometrical diagrams.

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

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.

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.

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.

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.

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?

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

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

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.

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?

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

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

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. . . .

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

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.

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...

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

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

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.

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

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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}.

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.

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 a rule which connects consecutive triangular numbers?

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

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

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

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

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

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

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

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 ...?

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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

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. . . .