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.

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.

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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?

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

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.

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

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

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

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

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

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.

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.

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

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

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

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.

Fourth 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.

A brief video explaining the idea of a mathematical knot.

Fifth 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!

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. . . .

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.

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

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. . . .

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

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

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

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

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

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?

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.

This problem opens a major sequence of activities on the mathematics of 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.

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

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. . . .

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.

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?

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

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.

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

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