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.

This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.

Can you sketch these difficult curves, which have uses in mathematical modelling?

Match the charts of these functions to the charts of their integrals.

Can you construct a cubic equation with a certain distance between its turning points?

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

Invent scenarios which would give rise to these probability density functions.

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

Why MUST these statistical statements probably be at least a little bit wrong?

This problem explores the biology behind Rudolph's glowing red nose.

How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

Here are several equations from real life. Can you work out which measurements are possible from each equation?

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

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

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

Can you make matrices which will fix one lucky vector and crush another to zero?

Explore the meaning of the scalar and vector cross products and see how the two are related.

Which line graph, equations and physical processes go together?

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

Use vectors and matrices to explore the symmetries of crystals.

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

Can Jo make a gym bag for her trainers from the piece of fabric she has?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

Explore the shape of a square after it is transformed by the action of a matrix.

Formulate and investigate a simple mathematical model for the design of a table mat.

Can you work out which processes are represented by the graphs?

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

Simple models which help us to investigate how epidemics grow and die out.

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

How would you design the tiering of seats in a stadium so that all spectators have a good view?

Get some practice using big and small numbers in chemistry.

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

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

Explore the properties of matrix transformations with these 10 stimulating questions.

Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

Which dilutions can you make using only 10ml pipettes?

Go on a vector walk and determine which points on the walk are closest to the origin.

Look at the advanced way of viewing sin and cos through their power series.

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.