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

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?

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

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?

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

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. . . .

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

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

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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

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.

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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

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

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 see how to build a harmonic triangle? Can you work out the next two rows?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

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

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

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

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. . . .

Explore the effect of reflecting in two parallel mirror lines.

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

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

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Explore the effect of combining enlargements.

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

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

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.

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

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?

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?

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?

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

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. . . .

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?

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

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

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

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?

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?

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

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

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...

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

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