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.

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

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?

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

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?

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?

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

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?

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.

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?

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. . . .

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?

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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

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.

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

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

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

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

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?

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

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

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

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

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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

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?

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?

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?

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

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

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

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 ?

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. . . .

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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?

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?

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

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

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

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

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

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

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?

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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