Can all unit fractions be written as the sum of two unit fractions?
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?
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.
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.
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?
A jigsaw where pieces only go together if the fractions are equivalent.
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.
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?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Can you find the area of a parallelogram defined by two vectors?
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. . . .
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Can you maximise the area available to a grazing goat?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Explore the effect of combining enlargements.
A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?
Explore the effect of reflecting in two parallel mirror lines.
Can you describe this route to infinity? Where will the arrows take you next?
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. . . .
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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 guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
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?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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 notice anything about the solutions when you add and/or subtract consecutive negative numbers?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
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?
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?
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.
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?