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. . . .

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

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. . . .

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?

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

A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45. . . .

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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

Can you break down this conversion process into logical steps?

Which dilutions can you make using only 10ml pipettes?

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

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

Making a scale model of the solar system

When Charlie retires, he's looking forward to the quiet life, whereas Alison wants a busy and exciting retirement. Can you advise them on where they should go?

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

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

My recipe is for 12 cakes - how do I change it if I want to make a different number of cakes?

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. . . .

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.

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the. . . .

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

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?

Practise your skills of proportional reasoning with this interactive haemocytometer.

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.

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

Build a scaffold out of drinking-straws to support a cup of water

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?

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

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?