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

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.

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

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

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

Can you prove that twice the sum of two squares always gives the sum of two squares?

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

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.

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

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.

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.

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

Derive an equation which describes satellite dynamics.

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

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

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.

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?

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?

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

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.

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

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}.

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

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

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

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

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.

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.

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

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?

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

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

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

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

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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?

If you know the perimeter of a right angled triangle, what can you say about the area?

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

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

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 ?

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

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

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

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

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. . . .

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.