Explore the effect of reflecting in two parallel mirror lines.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 describe this route to infinity? Where will the arrows take you next?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
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?
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.
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?
Can all unit fractions be written as the sum of two unit fractions?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?
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?
If you move the tiles around, can you make squares with different coloured edges?
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?
Explore the effect of combining enlargements.
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...
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?
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...
Can you maximise the area available to a grazing goat?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?
Can you find the area of a parallelogram defined by two vectors?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
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. . . .
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Is there an efficient way to work out how many factors a large number has?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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. . . .
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
How many different symmetrical shapes can you make by shading triangles or squares?
A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
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?
Which set of numbers that add to 10 have the largest product?