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

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

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

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

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

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?

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

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

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 ?

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.

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?

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?

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

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

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.

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

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

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

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

If you move the tiles around, can you make squares with different coloured edges?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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?

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?

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

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

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

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?

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

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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

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?

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

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.

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?

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?

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

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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

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

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

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. . . .