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

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.

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.

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?

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?

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

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

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?

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

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

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

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.

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

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

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

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?

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

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. . . .

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

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

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.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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.

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

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 ?

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

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

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

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?

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?

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

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

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

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?

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

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

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

The clues for this Sudoku are the product of the numbers in adjacent squares.

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?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?