An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.

Where should runners start the 200m race so that they have all run the same distance by the finish?

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

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.

Can Jo make a gym bag for her trainers from the piece of fabric she has?

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

How would you design the tiering of seats in a stadium so that all spectators have a good view?

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

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

Can you draw the height-time chart as this complicated vessel fills with water?

Starting with two basic vector steps, which destinations can you reach on a vector walk?

Use trigonometry to determine whether solar eclipses on earth can be perfect.

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

Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

Explore the shape of a square after it is transformed by the action of a matrix.

Explore the properties of matrix transformations with these 10 stimulating questions.

Go on a vector walk and determine which points on the walk are closest to the origin.

Can you make matrices which will fix one lucky vector and crush another to zero?

Is it really greener to go on the bus, or to buy local?

Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

Can you sketch these difficult curves, which have uses in mathematical modelling?

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

Use vectors and matrices to explore the symmetries of crystals.

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

Explore the meaning of the scalar and vector cross products and see how the two are related.

Get some practice using big and small numbers in chemistry.

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Which dilutions can you make using only 10ml pipettes?

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

Looking at small values of functions. Motivating the existence of the Taylor expansion.

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.

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. . . .

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

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

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

Can you work out which processes are represented by the graphs?

In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

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.

Make an accurate diagram of the solar system and explore the concept of a grand conjunction.

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Have you ever wondered what it would be like to race against Usain Bolt?