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

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

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!

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

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?

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

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

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

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. . . .

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.

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

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. . . .

How efficiently can various flat shapes be fitted together?

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

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.

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

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

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

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

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.

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. . . .

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

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.

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?

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

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

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

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.

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?

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.

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

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.

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

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

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.

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

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

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.

Investigations and activities for you to enjoy on pattern in nature.

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?

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