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 ?

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .

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

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?

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

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?

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. . . .

How many solutions can you find to this sum? Each of the different letters stands for a different number.

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.

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

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

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.

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

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?

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?

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

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?

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

Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this number. . . .

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

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

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

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. . . .

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

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

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

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?

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 find the area of a parallelogram defined by two vectors?

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

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?

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

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

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

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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

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.

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?

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.

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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

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?

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?

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