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?

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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

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. . . .

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

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?

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?

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

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

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

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.

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?

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

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

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. . . .

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?

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

Is there an efficient way to work out how many factors a large number has?

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

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?

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

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

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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

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

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

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?

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

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

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

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?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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

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?

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?

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

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?

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

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.