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.

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

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.

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. . . .

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?

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

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

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

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.

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

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?

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

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

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

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?

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

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

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

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

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

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?

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?

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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

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

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?

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

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

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

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?

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

Explore the effect of combining enlargements.

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

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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.

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

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

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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?

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.

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?

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?