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

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

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?

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

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

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

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.

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

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

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.

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?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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

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

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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?

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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.

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Can you describe this route to infinity? Where will the arrows take you next?

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?

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?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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".

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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

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

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

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

In a three-dimensional version of noughts and crosses, how many winning lines 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?

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

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

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

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?

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. . . .