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...

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. . . .

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?

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

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

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

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.

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

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

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?

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

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?

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

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

Explore the effect of reflecting in two parallel mirror lines.

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

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 combining enlargements.

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?

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

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

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.

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

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

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?

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.

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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 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 ?

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

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

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?

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?

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?

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

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

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

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

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

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?

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

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?

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.