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

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

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

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.

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?

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

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

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 ?

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. . . .

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?

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

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 ?

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. . . .

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 ?

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

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

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

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

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

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?

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?

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

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

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.

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?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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

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?

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?

Use the differences to find the solution to this Sudoku.

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

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.

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

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

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?

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

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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?

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

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.

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

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?

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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

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

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

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