Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to. . . .

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

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

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!

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

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.

At what positions and speeds can the bomb be dropped to destroy the dam?

Second in our series of problems on population dynamics for advanced students.

First in our series of problems on population dynamics for advanced students.

An advanced mathematical exploration supporting our series of articles on population dynamics for advanced students.

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

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

In this article for teachers, Alan Parr looks at ways that mathematics teaching and learning can start from the useful and interesting things can we do with the subject, including. . . .

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

This is the section of stemNRICH devoted to the advanced applied mathematics underlying the study of the sciences at higher levels

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

A car is travelling along a dual carriageway at constant speed. Every 3 minutes a bus passes going in the opposite direction, while every 6 minutes a bus passes the car travelling in the same. . . .

Fancy a game of cricket? Here is a mathematical version you can play indoors without breaking any windows.

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

This is about a fiendishly difficult jigsaw and how to solve it using a computer program.

Third in our series of problems on population dynamics for advanced students.

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?

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

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 article for students introduces the idea of naming knots using numbers. You'll need some paper and something to write with handy!

A brief video explaining the idea of a mathematical knot.

The third installment in our series on the shape of astronomical systems, this article explores galaxies and the universe beyond our solar system.

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. . . .

An account of how mathematics is used in computer games including geometry, vectors, transformations, 3D graphics, graph theory and simulations.

Fourth in our series of problems on population dynamics for advanced students.

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

Fifth in our series of problems on population dynamics for advanced students.

Edward Wallace based his A Level Statistics Project on The Mean Game. Each picks 2 numbers. The winner is the player who picks a number closest to the mean of all the numbers picked.

An advanced mathematical exploration supporting our series of articles on population dynamics for advanced students.

Sixth in our series of problems on population dynamics for advanced students.

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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

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

See how the motion of the simple pendulum is not-so-simple after all.

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

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

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

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

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.

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

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