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 ?

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

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?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

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

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?

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

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 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?

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.

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?

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

If a sum invested gains 10% each year how long before it has doubled its value?

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

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

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

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

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?

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

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

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

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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

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. . . .

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

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. . . .

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

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?

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?

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

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. . . .

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

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?

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?

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?

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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