This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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.

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

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

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?

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?

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?

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?

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

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. . . .

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

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

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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?

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

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

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

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

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

What is the same and what is different about these circle questions? What connections can you make?

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

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 ?

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

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

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?

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?

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

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

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

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?

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

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?

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?

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?