What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

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

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

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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.

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

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?

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?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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

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. . . .

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

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

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

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 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}.

What is the total number of squares that can be made on a 5 by 5 geoboard?

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

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

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.

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 =

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.

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. . . .

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. . . .

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. 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. . . .

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

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

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.

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

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

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

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

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?