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

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

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

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?

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

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.

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?

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?

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.

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...

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

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

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

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

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?

What can you say about the child who will be first on the playground tomorrow morning at breaktime in your school?

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.

Find at least one way to put in some operation signs (+ - x รท) to make these digits come to 100.

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?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

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

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?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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

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.

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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?

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

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

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?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

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

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

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?