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 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?

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?

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

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.

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

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?

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

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

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?

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?

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

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 ?

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

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.

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?

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

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

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?

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

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...

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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

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?

Explore the effect of combining enlargements.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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?

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?

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?

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

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.

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.

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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

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

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

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

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.

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?

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