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.

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

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

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

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?

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.

Derive an equation which describes satellite dynamics.

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

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

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

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.

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

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.

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.

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.

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.

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

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

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?

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

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...

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?

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

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

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

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

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

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

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.

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

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.

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

There are unexpected discoveries to be made about square numbers...

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

What is special about the difference between squares of numbers adjacent to multiples of three?

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?

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.

Can you explain what is going on in these puzzling number tricks?

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

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

Find the five distinct digits N, R, I, C and H in the following nomogram

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.

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Relate these algebraic expressions to geometrical diagrams.