Mike and Ellie find a code on a piece of paper hidden in one of the drums of the replica Bombe. The are some red numbers around the edge of the image, but these won't be important until chapter 6. The code itself consists of a 9 x 9 grid of letters that, at first sight, appear to make no sense. If you read each column upwards starting from the bottom left corner, however, then the hidden message is revealed:

I am the second of four. The rest will ensure. Solve my mystery to find my theory of psychohistory. AMT.

The 17th word is "theory".

This is an example of a tranposition cipher. The letters of the messages have not been replaced by other symbols, but have been rearranged or transposed.

Ellie, Mike and Barquith find a series of scratches on the bottom of a wooden box, supposedly a gift from Alan Turing. The scratches consist of low marks _, high marks - and slashes /. The marks are standard Morse code : the low mark represents a dash; the high mark represents a dot; and the slash is used to separate the letters. There is also a space between words of the form / /, which makes it a little easier to work out what is going on. The message is

You have the first my dear Barquith. The next in the Bombe joins its friends. The third is below SJ976695 and will reveal where our mystery ends.

The 19th word is "below". Morse code was developed to send messages using electrical telegraphs in order to convert language into electrical signals. The only possible signals that could be transmitted in the system were on and off. Hence, each letter of the alphabet (and each digit) was given a unique representation as a sequence of long and short pulses. Incidentally, as you probably know, modern communication networks still use simple on-off signals to encode the data, but these are collected into rather larger "packets".

The next clue was hidden a Bombe drum given to Barquith and, again ignoring the red numbers, consists of a ring of letters, with some parts missing, surrounding a list of positive and negative numbers. The numbers represent the number of steps around the ring from the present letter to the next letter either clockwise (positive) or anticlockwise (negative). Once you've worked that out, the only question is what's the starting letter. That requires a bit of trial and error, but actually it's the obvious starting letter: A. Counting on clockwise 12 letters from A gives I as the first letter and then counting back anticlockwise gives A as the second letter and so on. The entire message is

I am the first and have little to say. merely the start of this game. Study the box to find the next clue, persevering will bring you much fame.

The 24th word is "persevering".

Mike and Ellie dig up the next clue in the Peak District and it consists of a collection of punctuation marks. In fact the code is a simple substitution cipher that can be easily cracked with frequency analysis. The fact that there are still spaces between the words helps a lot. The deciphered message is

I am the third and here is my tale. You will find the fourth in the canal of Rochdale. Under the silt and the sludge and the slime, I rest beneath the muck and the grime. In Manchester town near a nice spot where I like to sit on a bench quite a lot.

The 19th word is "Rochdale".

Noma's computer has been hacked and the screen shows the same repeating video code. Each frame shows a three-tuple, three numbers in an ordered list e.g. (1,2,3), in white and a 5 x 5 grid of green letters and numbers that appear to be changing randomly. If you watch carefully, you will see that the green character in bottom right-hand corner is always a number and counts up from 0 to 9. Now, the first two of the white numbers only take values between 0 and 4, whereas the last one takes values between 0 and 9. In fact, the white numbers are coordinates indicating a particular position in the green grid. The first white number indicates the row (counting from the top and starting at zero); the second white number indicates the column (counting from the left and staring at zero); and the final white number indicates the time (i.e. the value of the number in bottom right corner of the grid). Once you've worked that out, you simply need to read off the solution:

The password is anatidae.

Anatidae is the name of family of water birds including wild geese. (A clue that all may not be as it seems with this particular treasure hunt.) This code was a lot of fun to create. If you look carefully you will see that the symbols in the green grid change more rapidly than the numbers in the bottom right corner, apart from the positions that correspond to letters in the code for the given time. The other symbols change randomly, but not always every frame. This gives the illusion of all the symbols changing all the time, when in fact there is quite a bit of structure.

The final code was fished out of the Rochdale canal and consists of 65 seemingly random letters as well as the familiar red numbers. The key to this code is the red numbers that run around the edge of the codes from chapters 1, 3, 4 and 6, as well as the extra torn scrap of paper shown in chapter 6. All these different scraps of paper fit together and if you count carefully you will see that there are 65 red three-digit numbers around the outside of the complete composite image. In other words, there is one number for each letter in the coded message. This is an example of a one-time pad where the ciphertext and the key (the red digits) need to be combined to decrypt the code. Each number tells you by how much to shift each letter in the code, but there is an added twist ... the numbers are in ternary (base 3). Normally, we work in base ten in which for a three digit number the same digits in the different columns indicate hundreds (ten squared), tens and ones. In base 3, the digits represent 9s (three squared), 3s and ones. Thus, 100 in base three is actually the number 9 and 011 would be 4. Starting in the bottom left hand corner of the complete page and reading clockwise the list of numbers is

022 100 201 021 200 012 001 202 012 201 202 010 120 112 202 200 100 002 210 202 100 120 112 201 212 012 200 012 100 112 202 022 012 020 100 012 110 011 120 020 121 201 221 010 022 120 022 100 201 202 120 200 221 201 012 110 011 120 112 020 120 210 112 011 202

Easter egg alert: if you convert these into letters using the standard code 1=A, etc, then you will find that they form a partial quote from Foundation by Isaac Asimov

his greatest contributions were in the field of psychohistory seldon found t

In order to decrypt the current code we simply count on the number of letters given by the corresponding red number, going back to A and starting again if the number takes us beyond Z. The first letter of the ciphertext is O, the first number is 022 (ternary), which is (2 x 3) + 2 = 8 in decimal. Counting on eight letters from O gives W, which is the first letter of the plaintext. Doing this for each letter in turn reveals the message

welldonethisistheendofthetraildiditamuseyouiknewyouwouldntfailmrb

which was the solution. Adding spaces and punctuation we have

Well done! This is the end of the trail. Did it amuse you? I knew you wouldn't fail. Mr B.

A final code is found in the casing of the drum fished out of the canal. After the first line, it consists of the letters R and C and the numbers 1 to 10. The first part of the first line QERTYUIOP are the letters than run across the first letter row of a standard UK keyboard, which is the clue to the method of encryption. The letter R stands for Row and the letter C stands for Column. The numbers tell you where the letter is located on a standard UK keyboard. For example R1C1 is the first letter on the first row: Q. If you spot that a group of (mostly four) characters represents each letter you can do frequency analysis as well. With your keyboard in front of you, it's straightforward to decode the message and find the final awful poem:

The ciphers were quite diabolical and the story was frankly illogical, but please do not fear we will return next year with brand new codes cryptological.

The final word is "cryptological".

Alan Turing Cryptography Competition 2020 is organised by the The Department of Mathematics at The University of Manchester.
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