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

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.

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

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.

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

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

How would you go about estimating populations of dolphins?

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

Get further into power series using the fascinating Bessel's equation.

Was it possible that this dangerous driving penalty was issued in error?

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

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?

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

Here are several equations from real life. Can you work out which measurements are possible from each equation?

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

Which line graph, equations and physical processes go together?

Get some practice using big and small numbers in chemistry.

Work out the numerical values for these physical quantities.

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

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

Use vectors and matrices to explore the symmetries of crystals.

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

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

Explore the relationship between resistance and temperature

This problem explores the biology behind Rudolph's glowing red nose.

Analyse these beautiful biological images and attempt to rank them in size order.

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

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

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

Build up the concept of the Taylor series

Which dilutions can you make using only 10ml pipettes?

Match the descriptions of physical processes to these differential equations.

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

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

Look at the advanced way of viewing sin and cos through their power series.

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

Work with numbers big and small to estimate and calulate various quantities in biological contexts.

When you change the units, do the numbers get bigger or smaller?

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

Work with numbers big and small to estimate and calculate various quantities in physical contexts.