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?

Can you deduce which Olympic athletics events are represented by the graphs?

Which countries have the most naturally athletic populations?

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

Invent a scoring system for a 'guess the weight' competition.

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

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?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

How do you choose your planting levels to minimise the total loss at harvest time?

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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

How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?

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?

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

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

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

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

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

Match the descriptions of physical processes to these differential equations.

Is there a temperature at which Celsius and Fahrenheit readings are the same?

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?

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

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

Is it cheaper to cook a meal from scratch or to buy a ready meal? What difference does the number of people you're cooking for make?

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

What shape would fit your pens and pencils best? How can you make it?

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

These Olympic quantities have been jumbled up! Can you put them back together again?

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

Make your own pinhole camera for safe observation of the sun, and find out how it works.

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

The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?

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

Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

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

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

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

If I don't have the size of cake tin specified in my recipe, will the size I do have be OK?

Explore the relationship between resistance and temperature

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

Get some practice using big and small numbers in chemistry.

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

Can you sketch graphs to show how the height of water changes in different containers as they are filled?

Examine these estimates. Do they sound about right?