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. . . .

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?

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?

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

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

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

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

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?

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?

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?

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

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

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

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

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

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

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?

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

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?

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

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?

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?

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

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.

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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

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

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 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. . . .

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

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

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?

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. . . .

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...

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?