Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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?

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?

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?

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?

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.

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

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?

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

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.

Can you work out the equations of the trig graphs I used to make my pattern?

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. . . .

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

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 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 ?

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?

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".

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

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

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?

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

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

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

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

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?

There are many different methods to solve this geometrical problem - how many can you find?

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

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. . . .

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

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

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

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.

If a sum invested gains 10% each year how long before it has doubled its value?

Which set of numbers that add to 10 have the largest product?

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

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

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

Use the differences to find the solution to this Sudoku.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

The clues for this Sudoku are the product of the numbers in adjacent squares.

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.

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?