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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
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?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Can you describe this route to infinity? Where will the arrows take you next?
Can all unit fractions be written as the sum of two unit fractions?
Explore the effect of combining enlargements.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 find the area of a parallelogram defined by two vectors?
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?
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Explore the effect of reflecting in two parallel mirror lines.
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?
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.
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
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?
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?
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.
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. . . .
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.
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
If you move the tiles around, can you make squares with different coloured edges?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
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?
Can you maximise the area available to a grazing goat?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?