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

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

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?

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?

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?

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?

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...

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

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

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

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?

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?

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.

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?

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...

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?

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

Explore the effect of combining enlargements.

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

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. . . .

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?

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

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?

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

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?

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

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?

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

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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?

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?

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

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.

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