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.

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.

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.

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

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

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

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

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

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

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. . . .

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

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.

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

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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

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

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?

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?

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?

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

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?

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

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?

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.

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

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

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

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 ?

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

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

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?

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

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?

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

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.

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?

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?

The clues for this Sudoku are the product of the numbers in adjacent squares.

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

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

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.