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

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

Can all unit fractions be written as the sum of two unit fractions?

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?

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

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

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

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

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

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

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?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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?

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 ?

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

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

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 ?

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

Use the differences to find the solution to this Sudoku.

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 ?

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

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

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?

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

A jigsaw where pieces only go together if the fractions are equivalent.

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

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

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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?

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

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.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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. . . .

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

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

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?

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?

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