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

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

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

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?

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.

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient 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.

Explore the effect of reflecting in two parallel mirror lines.

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. . . .

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 area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

What is the same and what is different about these circle questions? What connections can you make?

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?

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

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

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?

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

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.

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

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

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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?

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

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

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

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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

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

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

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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?

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?

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

Explore the effect of combining enlargements.

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. . . .

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

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

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

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?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?