Can you describe this route to infinity? Where will the arrows take you next?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Explore the effect of reflecting in two parallel mirror lines.
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 find an efficient method to work out how many handshakes there would be if hundreds of people met?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Explore the effect of combining enlargements.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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 you maximise the area available to a grazing goat?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
If you move the tiles around, can you make squares with different coloured edges?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
Can all unit fractions be written as the sum of two unit fractions?
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
Which set of numbers that add to 10 have the largest product?
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 different symmetrical shapes can you make by shading triangles or squares?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
If a sum invested gains 10% each year how long before it has doubled its value?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?