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. . . .

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

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?

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.

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!

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

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

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

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

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.

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

How do these modelling assumption affect the solutions?

An article demonstrating mathematically how various physical modelling assumptions affect the solution to the seemingly simple problem of the projectile.

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

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

This article explains the concepts involved in scientific mathematical computing. It will be very useful and interesting to anyone interested in computer programming or mathematics.

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

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

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. . . .

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. . . .

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

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

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

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

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. . . .

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

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

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

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

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

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

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.

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

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

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.

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?

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?

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

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

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

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

Can you make sense of information about trees in order to maximise the profits of a forestry company?

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

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

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

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

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

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