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

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

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

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

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?

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?

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?

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

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

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

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

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

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

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 find the area of a parallelogram defined by two vectors?

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. . . .

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

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

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

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

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 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?

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...

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.

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

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

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?

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

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

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?

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?

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

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

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.

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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.

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 you move the tiles around, can you make squares with different coloured edges?

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

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

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.

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