How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

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.

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

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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

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

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

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?

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

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

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

If you move the tiles around, can you make squares with different coloured edges?

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

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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

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

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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?

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

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. . . .

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?

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.

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

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?

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?

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

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?

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.

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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?

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

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

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.

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?

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

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

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?

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

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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

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.

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?