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.

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

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

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?

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

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?

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?

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

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?

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?

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

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

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...

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?

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?

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?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

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

Explore the effect of reflecting in two parallel mirror lines.

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

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

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?

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?

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?

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.

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.

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

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

Is there an efficient way to work out how many factors a large number has?

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

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

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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 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?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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

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?

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

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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