You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two. . . .

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!

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

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

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?

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

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

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. . . .

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?

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

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

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

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

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

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

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

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. . . .

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

Look at the calculus behind the simple act of a car going over a step.

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

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

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

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

The second in a series of articles on visualising and modelling shapes in the history of astronomy.

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. . . .

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

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. . . .

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?

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

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

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

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.

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

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

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

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

This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.

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

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

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.

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

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

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

How do scores on dice and factors of polynomials relate to each other?