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

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?

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

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

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

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

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

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?

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?

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 ?

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

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?

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

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

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.

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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

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?

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?

Explore the effect of combining enlargements.

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

Which set of numbers that add to 10 have the largest product?

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

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

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 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?

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?

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

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

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.

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

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

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?

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?

The clues for this Sudoku are the product of the numbers in adjacent squares.

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.

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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

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

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

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

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