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. . . .

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

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 ?

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?

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

Use the differences to find the solution to this Sudoku.

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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

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

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

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

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

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

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

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

There are lots of different methods to find out what the shapes are worth - how many can you find?

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?

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

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 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?

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

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?

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

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

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

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?

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?

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".

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

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

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.

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

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

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

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

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.

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

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

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?

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.