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

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?

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.

Kyle and his teacher disagree about his test score - who is right?

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?

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.

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

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.

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. . . .

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

Relate these algebraic expressions to geometrical diagrams.

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.

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.

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

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 explain why a sequence of operations always gives you perfect squares?

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

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}.

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.

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

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

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?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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

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.

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.

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

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

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

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.

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

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

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?

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.

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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

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

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 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?

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

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

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

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?