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

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?

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

What does the empirical formula of this mixture of iron oxides tell you about its consituents?

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

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

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

Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?

Which dilutions can you make using only 10ml pipettes?

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.

Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions

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

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 Pythagoreans noticed that nice simple ratios of string length made nice sounds together.

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 work out the fraction of the original triangle that is covered by the green triangle?

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

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?

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. . . .

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

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?

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

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

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.

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

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.

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

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

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

Move the point P to see how P' moves. Then use your insights to calculate a missing length.

Can you find an efficent way to mix paints in any ratio?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

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

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. . . .

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?