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

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

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

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?

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

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

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

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?

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?

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

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

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

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?

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

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.

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?

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.

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

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

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

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?

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

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 ?

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?

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?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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

Explore the effect of reflecting in two parallel mirror lines.

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

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

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. . . .

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

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?

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

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?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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?

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 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?

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

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

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?

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

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