Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the. . . .

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

How much peel does an apple have?

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

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

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

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

What's the most efficient proportion for a 1 litre tin of paint?

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

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

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 is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

If a ball is rolled into the corner of a room how far is its centre from the corner?

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?