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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.

Procedure Solver

Can you think like a computer and work out what this flow diagram does?


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

Josip from Australia
Joseph from Hong Kong
(and probably also Patrick from Woodbridge School and maybe others - I accidentally forgot to leave the submit a solution tab open!)


Josip give the following nice advice


Ok, so I recently started going some cryptography in my spare time after
entering a YouTube competition. After a few hours, I finally caught on the
way to step-by step solve the puzzles. For this particular puzzle, I got
the hint to use frequency analysis from the previous puzzle. Instead of
taking a trip down the tedious path and going through and looking at the
repetition of each letter and word pattern, I used this handy little
website. To compliment me with
switching letters back and forth, I used which also contains a
"FA", but makes it much easier to replace letters and see
immediate results. (I always compare frequency in the text to frequency in
theory and try to match them.)

The following is an example I use to step my way to more efficiently
cracking the code; efficiency is everything. This gave me the vital letter
frequency, as well as 2,3,4,5 and 6 letter repetitions! I immediately
looked at the 3 letter repetition section to notice "qao". I
automatically assumed this to be the word "the", as it is many
times. Now that I have those three letters, I apply this to the six letter
section in hope to get something out. The "goqqon" word gets my
attention because it contains two pairs of like letters. Also because we
know that o=e, it significantly helps. Now, I thought to myself in the
context of the previous two puzzles, 'what could _eXXe_' be? (with XX being
a double letter). It didn't take long for me to think of 'letter'!! I also
noticed that it was able to be plurals, so I added an 's'. From here,
things got slightly easier. With the help of frequency analysis, I tried to
strip the less common letters and replace them with more popular ones, and
sometimes with luck. Research in word patterns I thought benefited me
hugely, and so I recommend them too;

I started strongly from around the centre of the puzzle, with "ALL THE
LETTERS IN A MESSAGE IS THAT" and then slowly edged my way to find
smaller words in between that made things clearer. As I write my
'confirmed' (used) letters down on paper, I have a remaining list which can
still be rearranged anywhere without affecting the already made words. This
is really good for times when you are stuck, (as I was). You can simply
change a few letters and hopefully see a pattern that makes you think.
Sometimes you can try to fill in the sentence with your own words also. e.g
"THIS IS CALLED A _____ _____". I knew from the context that the
puzzle was referring to another cipher (which also gave away the word
'cipher') but quickly researched a few cipher names to reveal a nice fit of
'vigenere'. The other tricky part was arranging the less common letters.
However, sometimes the surrounding letters gives it away, such as
"freQuency". Also, when you have words already made, spacing them
helps significantly.

The solution is:

The last message was encrypted using an affine cipher, 'a' to 'q', 'b' to 'z', 'c' to 'i', i.e. moving on nine letters each time. The problem with just using one cipher to encrypt all the letters in a message is that we maintain the frequency of letters in the plaintext, so we're able to use frequency analysis to help decipher it. We could instead use different alphabets for different parts of a message. For example we could encrypt the odd letters with one cipher and the even letters with another. This is called a Vigenere cipher with keyword length two. Good luck!

keyword cipher: