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

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

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

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?

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?

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

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?

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

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.

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?

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. . . .

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

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

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?

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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.

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?

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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.

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. . . .

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.

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?

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

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.

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?

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}.

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

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

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

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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.

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

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 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.

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

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?

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

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

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