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.

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

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

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.

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?

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.

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.

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 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.

If you know the perimeter of a right angled triangle, what can you say about the area?

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

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?

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...

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

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?

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?

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

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

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

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.

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 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.

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

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

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

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

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

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

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

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.

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

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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

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}.

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

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

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?

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

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

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

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