between the two 1s there is one digit giving 1?1,
between the two 2s there are two digits giving 2??2,
between the two 3s there are three digits giving 3???3.
It is not difficult to put a list of decimals in order of size. But what about ordering fractions?
John Farey introduced sequences of sequences of fractions in order of size, now called Farey sequences. Can you discover his method from the following examples?
What patterns do you see in them?
Suppose you have to share 9872 gold coins between 8 people.
You give the coins out in thousands until there are less than 8000 left.
Then you give out one hundred at a time until there are less than 800, then […]
He labelled each bag with the number of coins inside it.
He could then pay any sum of money from 1 coin to 15 coins using one or more of the bags without opening any of the bags.
Give a proof linked to this picture of the formula $a^2 – b^2 = (a + b)(a – b)$ .
2. What do you notice about the difference between squares of consecutive numbers?
Compare the differences between squares of numbers that differ by 1 […]
This picture shows how you can fit 2 identical copies of a triangle together to form a rectangle.
Can you use it to find the number of dots in the fifth triangle number without counting the dots?
Can you find any […]
Is there more than one?
It is quite easy to find a few solutions. The big challenge is to find all possible solutions.
Can you provide a convincing argument that you have found […]
Explore the pattern you get when you carry on these multiplications up to 37 × 54.
Click here for MAGIC OF 37 worksheet.
Click here for the Notes […]
How many lines are there in your diagrams? How many lines are there in the original diagram? Can you find a way of working out […]
You might like to start by investigating the patterns you get when you shade multiples of a number on a grid. Download here
For the prime sieve investigation use a 100 square grid (download here):
Circle the number 2. Put a line through every multiple of 2 up to 100.