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

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

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.

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

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?

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.

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

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

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.

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

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

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?

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. . . .

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

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?

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?

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

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

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

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.

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

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?

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

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

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

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

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?

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?

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 ?

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

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

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

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

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

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?

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

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

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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?

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

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

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

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

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