An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

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

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

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 is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

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?

This problem opens a major sequence of activities on the mathematics of population dynamics for advanced students.

Work in groups to try to create the best approximations to these physical quantities.

In Classical times the Pythagorean philosophers believed that all things were made up from a specific number of tiny indivisible particles called ‘monads’. Each object contained. . . .

Is it really greener to go on the bus, or to buy local?

See how differential equations might be used to make a realistic model of a system containing predators and their prey.

How do decisions about scoring affect who wins a combined event such as the decathlon?

How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.

bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your. . . .

A look at the fluid mechanics questions that are raised by the Stonehenge 'bluestones'.

From the atomic masses recorded in a mass spectrometry analysis can you deduce the possible form of these compounds?

From the information you are asked to work out where the picture was taken. Is there too much information? How accurate can your answer be?

Financial markets mean the business of trading risk. The article describes in simple terms what is involved in this trading, the work people do and the figures for starting salaries.

Scientist Bryan Rickett has a vision of the future - and it is one in which self-parking cars prowl the tarmac plains, hunting down suitable parking spots and manoeuvring elegantly into them.

A simple method of defining the coefficients in the equations of chemical reactions with the help of a system of linear algebraic equations.

An example of a simple Public Key code, called the Knapsack Code is described in this article, alongside some information on its origins. A knowledge of modular arithmetic is useful.

Playing squash involves lots of mathematics. This article explores the mathematics of a squash match and how a knowledge of probability could influence the choices you make.

Noticing the regular movement of the Sun and the stars has led to a desire to measure time. This article for teachers and learners looks at the history of man's need to measure things.

In four years 2001 to 2004 Arsenal have been drawn against Chelsea in the FA cup and have beaten Chelsea every time. What was the probability of this? Lots of fractions in the calculations!

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

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

STEM students at university often encounter mathematical difficulties. This articles highlights the various content problems and the 7 key process problems encountered by STEM students.

An introduction to the ideas of public key cryptography using small numbers to explain the process. In practice the numbers used are too large to factorise in a reasonable time.

How can people be divided into groups fairly for events in the Paralympics, for school sports days, or for subject sets?

In this article, Alan Parr shares his experiences of the motivating effect sport can have on the learning of mathematics.

engNRICH is the area of the stemNRICH Advanced site devoted to the mathematics underlying the study of engineering

If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

How efficiently can various flat shapes be fitted together?

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

As part of Liverpool08 European Capital of Culture there were a huge number of events and displays. One of the art installations was called "Turning the Place Over". Can you find our how it works?

For teachers. Yet more school maths from long ago-interest and percentages.

chemNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of chemistry, designed to help develop the mathematics required to get the most from your study. . . .

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

PhysNRICH is the area of the StemNRICH site devoted to the mathematics underlying the study of physics