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?
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 ?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Explore the effect of reflecting in two parallel mirror lines.
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT 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. . . .
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?
Can you work out the dimensions of the three cubes?
How many different symmetrical shapes can you make by shading triangles or squares?
If a sum invested gains 10% each year how long before it has doubled its value?
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.
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
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 ?
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?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
Can you describe this route to infinity? Where will the arrows take you next?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you maximise the area available to a grazing goat?
If you move the tiles around, can you make squares with different coloured edges?
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.
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
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?
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 .
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?
Which of these games would you play to give yourself the best possible chance of winning a prize?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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".
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Can you find the area of a parallelogram defined by two vectors?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
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.
What is the same and what is different about these circle questions? What connections can you make?
Can you find rectangles where the value of the area is the same as the value of the perimeter?