# Turing's Treatise on the Enigma

repetitions between the message settings for the day in question and message settings of previously solved days. We might also do a 'Banburismus' i.e. we might make use of the fact that if two messages are written out with letters that were enciphered at the same position written in the same column, then the number of repetitions of letters in a column will be the same as if the messages had not been enciphered, and therefore will be greater on average than if the messages had been otherwise placed. Actually this effect was very small for the Naval traffic in 1937 and earlier. The repetition frequency was 1/20, as compared with 1/16.5 for the 1940 Naval traffic and the Air traffic, 1/12 for plain language German, and 1/26 for incorrectly placed enciphered messages (the repetition frequency is the ratio of the number of identical pairs in the columns to the total number of pairs in columns, identical or not). With so low a repetition frequency it is extremely difficult to equate the boxes unless the traffic is rather heavy. This method however applies quite well with the Air traffic up to Sept 14 1938, but there are better methods of equating. Once the boxes have been equated by one means or another we shall have many more cases of half-bombes which we can assume to have been unsteckered. This method will nearly always get the result, if the equating can be done.

After we have found the rod position of the Grundstellung and the Stecker it only remains to find the Ringstellung. Usually this would be known already, as, at this period, the wheel order and Ringstellung were only changed about once a fortnight. However if these have just been changed it is necessary to read one message. This could always be done, as a great many messages were sent in two or more parts. In such cases the call signs and signatures of the parts were essentially the same, and the parts after the first began by saying that they were continuations, giving the time group of the last part of the message as a reference. The method of giving numbers at that time was to use

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