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 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?

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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.

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

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

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?

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

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

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

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?

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?

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?

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

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

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.

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?

Explore the effect of reflecting in two parallel mirror lines.

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?

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

Explore the effect of combining enlargements.

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

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

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 ?

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

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

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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.

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

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?

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

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

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

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?

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

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

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

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

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. . . .

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?