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

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?

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

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

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?

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?

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.

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.

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

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. . . .

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

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?

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

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?

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

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

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

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?

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

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?

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

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

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?

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?

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

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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

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

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

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

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

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

Can all unit fractions be written as the sum of two unit fractions?

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.

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?

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?

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?

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.

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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

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?

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