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?

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 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.

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?

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...

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

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?

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.

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?

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.

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?

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

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?

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

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

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

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?

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

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. . . .

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

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?

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

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

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

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

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

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

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?

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

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 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 end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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?