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 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

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

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

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

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.

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

Explore the effect of reflecting in two parallel mirror lines.

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

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

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?

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

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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

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?

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?

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

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

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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 combining enlargements.

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?

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

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?

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

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

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?

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.

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.

How many different symmetrical shapes can you make by shading triangles or squares?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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?

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

A jigsaw where pieces only go together if the fractions are equivalent.

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

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

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

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

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

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?

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?

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

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?

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. . . .

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?