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

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

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

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?

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

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

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

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

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

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.

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?

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

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.

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

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...

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.

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

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

Explore the effect of combining enlargements.

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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

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?

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

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?

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

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

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

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?

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?

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.

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

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

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

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?

Use the differences to find the solution to this Sudoku.

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

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

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?

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?

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?

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

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

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

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