How did the cryptographers of Bletchley Park figure out the chi stream of the Lorenz cipher?

Question

How did the Bletchley Park code breakers figure out the chi stream of the Lorenz cipher, that was obscured in the ciphertext, which British code breakers eventually decoded? It's written in The Rutherford Journal that John Tiltman deduced the key using a depth found on 30 August 1941. Since he couldn't figure how the key was generated, it was given to W.T. Tutte, who figured out how the key was generated without ever looking at the Lorenz cipher machine.

Jack Copeland writes in Colossus: The Secrets of Bletchley Park's Codebreaking Computers (p.364)

Here was a method of wheel setting! Δ(χ1+χ2) was a periodic sequence, supposed known, with a period of 41×31=1271. Lay it against Δ(C1+C2) in each of the 1271 possible relative positions and count the number of agreements for each one.

It's obvious that the British code breakers had the ciphertexts in hand, from which they could obtain the delta-ed version of the sum of the first and second impulse of an intercepted ciphertext, i.e., Δ(C1+C2). I would like to know how the code breakers figured out the delta-ed version of the chi stream of the Lorenz cipher machine, i.e., Δ(χ1+χ2).

Answer 1

The answer can be found here:

The initial function of Colossus was to help determine the starting point of the wheels. Colossus read the cipher’s stream of characters and counted the frequency of each character. Cryptographers then compared the results to the frequency of letter distribution in the German language and to a sample chi-wheel combination, continually refining the chi-wheel settings until they found the optimal one.

The setting that gave the highest count of •s was found to be optimal (F. Carter, Codebreaking with the Colossus Computer, 2008, pp. 16-17).

They understood the settings of the wheels because of the extraordinary work of William Thomas Tutte, OC FRS FRSC. From Wikipedia:

At Bletchley Park, mark impulses were signified by x and space impulses by •. For example, the letter "H" would be coded as ••x•x. Tutte's derivation of the chi and psi components was made possible by the fact that dots were more likely than not to be followed by dots, and crosses more likely than not to be followed by crosses. This was a product of a weakness in the German key setting, which they later eliminated. Once Tutte had made this breakthrough, the rest of the Research Section joined in to study the other impulses, and it was established that the five chi wheels all advanced with each new character and that the five psi wheels all moved together under the control of two mu or "motor" wheels. Over the following two months, Tutte and other members of the Research Section worked out the complete logical structure of the machine, with its set of wheels bearing cams that could either be in a position (raised) that added x to the stream of key characters, or in the alternative position that added in •.