P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

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?

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

Can you fill in the mixed up numbers in this dilution calculation?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

Which dilutions can you make using only 10ml pipettes?

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

Four jewellers share their stock. Can you work out the relative values of their gems?

Which exact dilution ratios can you make using only 2 dilutions?

Which dilutions can you make using 10ml pipettes and 100ml measuring cylinders?

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. . . .

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

Do you have enough information to work out the area of the shaded quadrilateral?

One night two candles were lit. Can you work out how long each candle was originally?

In this question we push the pH formula to its theoretical limits.

Investigate some of the issues raised by Geiger and Marsden's famous scattering experiment in which they fired alpha particles at a sheet of gold.

Do each of these scenarios allow you fully to deduce the required facts about the reactants?

Ever wondered what it would be like to vaporise a diamond? Find out inside...

A brief introduction to PCR and restriction mapping, with relevant calculations...

Andy is desperate to reach John o'Groats first. Can you devise a winning race plan?

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

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