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?

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

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

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?

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

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...

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.

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?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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".

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?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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. . . .

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?

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?

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

If a sum invested gains 10% each year how long before it has doubled its value?

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?

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

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?

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.

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?

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

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

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?

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?

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

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?

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

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

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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?

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

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

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