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

Can you find the lap times of the two cyclists travelling at constant speeds?

Derive an equation which describes satellite dynamics.

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?

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

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

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

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

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

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?

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

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?

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

Can you find the area of a parallelogram defined by two vectors?

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

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?

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

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.

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

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 =

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?”

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. . . .

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.

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

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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.

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

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.

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?

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

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?

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

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 ?

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.

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

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?

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

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

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.

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

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