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.

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

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?

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

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

Get some practice using big and small numbers in chemistry.

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

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

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.

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

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

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

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

Which line graph, equations and physical processes go together?

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

Work out the numerical values for these physical quantities.

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

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

Which units would you choose best to fit these situations?

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

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.

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

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.

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

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

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

Build up the concept of the Taylor series

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

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

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

Explore the relationship between resistance and temperature

Which dilutions can you make using only 10ml pipettes?

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?

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

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

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

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

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

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

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

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