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.

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?

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.

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 of these games would you play to give yourself the best possible chance of winning a prize?

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

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

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. . . .

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

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

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.

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

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

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

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?

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

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. . . .

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.

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?

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

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

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

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.

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

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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

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 ?

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?

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

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?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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

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

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 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. . . .

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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

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

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

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

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

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

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

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?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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?