Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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

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. . . .

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

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.

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

If you move the tiles around, can you make squares with different coloured edges?

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

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?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

Explore the effect of combining enlargements.

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

What is the same and what is different about these circle questions? What connections can you make?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

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?

Can you find the area of a parallelogram defined by two vectors?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

How many winning lines can you make in a three-dimensional version of noughts and crosses?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

How many different symmetrical shapes can you make by shading triangles or squares?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

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 ?

Which of these games would you play to give yourself the best possible chance of winning a prize?