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.

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

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

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.

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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

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?

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?

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

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

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?

Is there an efficient way to work out how many factors a large number has?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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

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?

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

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

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.

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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.

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

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

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

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

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?

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?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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?

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. . . .

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

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

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?

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

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?