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?

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

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?

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

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

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?

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

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

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

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

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

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?

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?

Explore the effect of combining enlargements.

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.

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

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

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. . . .

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. . . .

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?

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

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

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

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?

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?

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?

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?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

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?

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

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

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

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?

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

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

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.

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

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