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

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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?

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

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

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

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

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?

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

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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...

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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

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

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?

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

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

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

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

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

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

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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

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

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

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

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?

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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.

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.

Can you maximise the area available to a grazing goat?

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

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?

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

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 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.

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?

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

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.

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?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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?

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

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?