### 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.

### Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

# A Fine Thing?

##### Stage: 5 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, Woodbridge School, England
Matthew from Wilson's School in England
Sebastian from QEH in Bristol, England
Joseph from Hong Kong

Well done code-breakers!

The solution is as follows:

The llama or Lama glama is a South American camelid, widely used as a pack and meat animal by Andean cultures since pre-hispanic times. Although early writers compared llamas to sheep, their similarity to the camel was soon recognized. They were included in the genus Camelus along with alpaca in the Systema Naturae of Linnaeus. Someone encrypting a message can make decryption more difficult by choosing unusual words and removing punctuation and the spacing between words. See if you can work out the method used for encrypting this message. Think about sequences if you get stuck.

Affine cipher:
a -> Q
b -> Z
c -> I
d -> R
e -> A
f -> J
g -> S
h -> B
i -> K
j -> T
k -> C
l -> L
m -> U
n -> D
o -> M
p -> V
q -> E
r -> N
s -> W
t -> F
u -> O
v -> X
w -> G
x -> P
y -> Y
z -> H

x->9x+8