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

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

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

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

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

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

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

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

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. . . .

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

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

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

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

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

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

Sixth 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.

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

How do these modelling assumption affect the solutions?

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

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!

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

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.

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. . . .

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

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

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

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

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. . . .

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

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

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.

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

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?

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

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

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

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

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

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

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.

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. . . .

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?

Explore the transformations and comment on what you find.

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

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

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.

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. . . .