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

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

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.

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

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.

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

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?

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

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

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

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

Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this number. . . .

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

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

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. . . .

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

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

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

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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?

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. . . .

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.

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?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

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

Explore the effect of combining enlargements.

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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?

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

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?

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 circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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 hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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

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

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

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

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?

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

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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?