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 size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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. . . .
Start with two numbers and generate a sequence where the next number is the mean of the last two 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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
How many different symmetrical shapes can you make by shading triangles or squares?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Here's a chance to work with large numbers...
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Explore the effect of reflecting in two parallel mirror lines.
Can you describe this route to infinity? Where will the arrows take you next?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 move the tiles around, can you make squares with different coloured edges?
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.
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Explore the effect of combining enlargements.
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?
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 ?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you maximise the area available to a grazing goat?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
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?
Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.
Is there an efficient way to work out how many factors a large number has?
A jigsaw where pieces only go together if the fractions are equivalent.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
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?
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.
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?