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?

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?

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

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?

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

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

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?

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?

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

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 ?

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

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

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

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

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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

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.

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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

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

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?

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. . . .

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...

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?

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

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

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

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.

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

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?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

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

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?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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.

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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