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even though the original chance that any particular set of settings is correct is as small as 1 in 41x31x29x26x23, that is 1 in 22 million.

Fortunately it is not necessary to try out every combination of chi settings individually. We can count the combined frequency of 0 and M say, without knowing (or bothering) where chi 3 is set, by counting the number of positions where ΔD1 is a dot, ΔD2 a dot, ΔD4 a cross, and ΔD5 a cross. Therefore using the combined counts of 0 and M and of others pairs of letters differing from each other only on the third impulse we can set chi 1, chi 2, chi 4, and chi 5, and then go back later to chi 3.

When attempting to set a message we normally start with the '1+2 BREAK IN'. We count the ΔD1 + ΔD2 = dot, that is the combined frequency of the letters /9HT OMN3 AUQW 58KJ, most of which occur frequently in ΔD. On a correct de-chi the count of these 16 letters should be well above the count of the other 16 letters for which ΔD1 + ΔD2 = x. On a de-chi at incorrect settings the combined frequency of any 16 letters should be about half the total number of letters counted. So by counting possible de-chis on the first two impulses only, at the 41x31 = 1271 possible settings for chi 1 and chi 2, we can probably set these wheels. Then, by counting the frequency of other suitably chosen sets of letters we can set the other chis in turn (either singly or in pairs). It is not necessary to set all chis simultaneously.

Even so, the counting of the 1271 possible Δde-chis on the first two impulses and other similar operations are not jobs which could be undertaken by hand. The COLOSSUS is a machine which has been devised to do these jobs at high speeds. It can be made to record the answers only at such settings as are likely to be correct. A ROBINSON is a more general machine which can be used for the same purpose.

If a transmission is too short then the correct ΔD count will not stand out sufficiently from the others to make the settings certain. When the language is moderately good the minimum lengths required are very roughly as shown in the following table (d is the dottage of μ_{37}).

15 | 18 | 21 | 24 | 27 | ||

Rough minimum | 6200 | 4000 | 2400 | 1700 | 1200 |

These figures have a very large probable error.

(e) __Chi Breaking__.

If there is a very strong ΔD count for a given transmission
it is possible not only to select the settings used for making the correct
ΔΧ stream if the wheel patterns are known, but to determine the
patterns of the wheels themselves if they are not known. This is equivalent
to selecting the correct ΔD count from the series of letter counts made
with ALL POSSIBLE wheel patterns, and can often be done even though the original
chance that any set of wheel patterns is correct is 1 in 2 to the power of
(41+31+29+26+23) = 1 in 2^{150} = 1 in 10^{45}. (in fact the
figure 10^{45} is an overstatement, as the Germans impose restrictions
on themselves in the choice of wheel patterns which reduce the figure to about
10^{38}. (See 25X)

The 1+2 RECTANGLE which is made on Colossus or Garbo and CONVERGED by hand is a means of finding the patterns of ΔΧ_{1} and ΔΧ_{2} which maximise the number of letters of ΔD in which ΔD1 + ΔD2 = dot. The extent to which this frequency can be made to exceed ½ when the optimum patterns have been chosen, determines (a) how much relation the optimum patterns are likely to have to those really used and (b) whether it is worth while to attempt to use the most reliable characters in the optimum pattern for setting other messages enciphered on the same wheel patterns or as a start for COLOSSUS WHEEL-BREAKING. In Colossus wheel-breaking attempts are made to find the deltaed patterns of all the chis which will lead to the strongest ΔD count.

Unless there is a transmission of over 4000 letters it is unlikely that the optimum ΔΧ_{1} and ΔΧ_{2} will be strong enough to be in any way significant and therefore chi-breaking by means of the rectangle will be impossible.

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