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

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.

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?

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

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?

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

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?

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

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.

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

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

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

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?

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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

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

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?

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

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?

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.

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

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?

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

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?

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

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

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

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?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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?

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

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...

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

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?

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.