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

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.

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

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

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. . . .

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

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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?

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

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

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?

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

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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

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

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.

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 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.

Can you make sense of these three proofs of Pythagoras' Theorem?

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.

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

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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

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.

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

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

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?

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.

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

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

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?

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

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

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

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

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

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?

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?

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 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?

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.

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .