### Route to Root

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.?

### Divided Differences

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.

### Geomlab

A geometry lab crafted in a functional programming language. Ported to Flash from the original java at web.comlab.ox.ac.uk/geomlab

# Jluuis or Even Asutguus?

##### Age 16 to 18 Challenge Level:

Have you managed to solve the entire Stage 5 Cipher Challenge? Solutions are now closed, but perhaps you want to take up the full challenge.

Successful solvers of this part were

Patrick from Woodbridge School, England
An Anonymous Solver from Somewhere in the US
Joseph from Hong Kong

The solution is:

This was a Caesar shift of seventeen followed by a transposition of rows and columns. This of course retains the letter frequencies of English, which probably helped you decipher this. In this case, the two methods of encryption commute, however this isn't always the case. Apart from some special cases, if we use a vigenere cipher and then a transposition, we will get a different result depending on which order we do them in. Can you find any cases for which these will commute?