A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if
you know any term xn, you can find the next term xn+1 using the
formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of
this sequence. What do you notice? Calculate a few more terms and
find the squares of the terms. Can you prove that the special
property you notice about this sequence will apply to all the later
terms of the sequence? Write down a formula to give an
approximation to the cube root of a number and test it for the cube
root of 3 and the cube root of 8. How many terms of the sequence do
you have to take before you get the cube root of 8 correct to as
many decimal places as your calculator will give? What happens when
you try this method for fourth roots or fifth roots etc.?
When in 1821 Charles Babbage invented the `Difference Engine' it
was intended to take over the work of making mathematical tables by
the techniques described in this article.
Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.
Take some numbered cards; between ten and twenty should be enough (you could use a suit from a pack of playing cards).
Shuffle the cards.
Then organise your deck of cards into numerical order.
What method did you use to put them in numerical order?
Can you think of any other ways you could have sorted them?
Here are some different sorting algorithms you could try. You may find it easiest to lay the cards out in a line to keep track of their order and see what's happening at a glance.
Try each algorithm a few times, and keep a record of how many 'moves' or 'swaps' you do. You could work with a friend and 'race' against each other to see who sorts their pack the quickest.
If you are struggling to make sense of the written algorithms, here are some videos showing each algorithm being performed.
Here are some questions to consider: