Another count was made for the fifth impulse of the first letter. This count
differed from all the others in that nearly all the entries in any one column were
the same. This showed that only the X wheels were effective in the encipherment
of the first letter.
Similar results were obtained for the first and third impulses. The other
two were avoided because they are the ones in which + and Z differ, so that these
two impulses would, it was thought, present more difficulty than the others.
The difficulty that had been presented by the first cipher letter in March
and April was now explained, and it was no longer a matter of complete
indifference whether the wheel patterns of a Tunny machine were reversed or not.
This property of the first letter was peculiar to ZS 40 ( the first model of the
German Tunny machine).
(d) Construction of pattern fragments.
On the assumption that almost all messages began with a group of +'s,
followed by a group of Z's it followed that nearly every message began in the
fifth impulse with a sequence of crosses. ( At least 6 crosses, to judge by the
March and April traffic). Since the PSI wheel did not operate in the first
place, the nature of the X character in the wheel-setting corresponding to each
X indicator could be determined from the count of the fifth impulse of the 1st
letter. Since each setting of the 23-wheel corresponds to some indicator letter,
the number of crosses in the pattern of the fifth X wheel could at once be
deduced. It was found to be 11. Of course the count or the first letter did
not suffice to determine the pattern of the wheel, since the wheel settings were
not in the order of the indicator letters.
The analysis of the count of the 2nd letter was more complicated since the
PSI wheels were now operative. Each cipher character was the sum of a clear
character assumed to be x, a X character fixed by the X indicator, and a PSI
character fixed by the PSI indicator. However, if a particular X character was
assumed to be dot, the values of a number of PSI characters could be deduced from
the row of the square corresponding to that X character. Then more X characters
could be deduced from these PSI characters, and so on. This process was carried
on until it terminated, and so sets of X and PSI characters were obtained.
Since these led to very little inconsistency, they were assumed to be the correct
ones. Some of the PSI characters were uncertain, since the corresponding
rows were almost empty, but all the X characters were obtained with a fair
certainty. The first assumption, that a particular X character was x, might
have been wrong: it would have then been necessary to reverse all the X and
PSI characters finally obtained. This point was settled by using the fact that
the number of crosses in the fifth X wheel was 11.
The count of the third letter was analysed in the same way. It was found
that the PSI wheel always moved on between the second and third letters.
We will now summarise the information which had been obtained at this stage.
We shall use the term "pattern-fragment for A" to denote a short sequence of dots
and crosses in a wheel beginning at the setting which, with the indicator
A, corresponds to the first letter of the message in the case of a X wheel, and to
the second letter in the case of a PSI wheel.
The pattern-fragments of the fifth X wheel were known to three places, and
the pattern fragments of the fifth PSI wheel were known to two places. A check
on the working was now possible, for by the nature of the PSI wheels the pattern