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.

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

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.

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

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 fill in the blanks in truth tables to record. . . .

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?

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.

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.

Third 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.

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

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

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.

A brief video explaining the idea of a mathematical knot.

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

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

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

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

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

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. . . .

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. . . .

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. . . .

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

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

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

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?

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

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.

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?

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!

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?

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.

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

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

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. . . .

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?

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

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

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

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