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.

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?

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

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?

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.

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

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

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

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

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

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

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?

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?

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?

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 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 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?

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

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

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

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

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. . . .

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

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

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?

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

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

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?

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?

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.

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

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

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?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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?

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

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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

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?