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

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

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

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

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.

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

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

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

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

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

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?

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

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

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

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

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?

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.

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?

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

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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?

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.

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Explore the effect of combining enlargements.

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?

Explore the effect of reflecting in two parallel mirror lines.

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?

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?

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?

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?

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?

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?

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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

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

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 arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

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

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

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

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

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

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

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

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

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.

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

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