Solving a short crib
The chief application of the Turing sheets is to the solution of cribs from a length of 2 to 6 letters. When the U.K.W. does not rotate, we set up the inverse rods as usual, but find that by no means all the incorrect positions are eliminated by coupling contradictions. We therefore look to see whether there is any position in which the couplings can occur. Take for example the crib ANX, with cipher BRX and wheel order I III II (red, green, purple), U.K.W. pos. 0. We set up the inverse rods as in Fig 57, and for each column of the resulting set up compare the lines of the catalogue named after the pairs in the column. For each pair we shall want to find quickly the right sheet on which to look, and this means subtracting the pair on the diagonal (i.e. finding their distance apart on qwertzu). To do this we can either have a table of differences or else use ‘mystic number rods’.
‘Mystic numbers’
Fig 58 shows a table of ‘mystic numbers’ for the red wheel. The meaning of the table is this. Take the 8th line for example. It could be made by taking inverse rod Q and inverse rod O, O being eight places on along qwertzu from Q. We lay the two rods together and find the differences of the resulting pairs: e.g. the fifth entry in line 8 is 6, and the fifth letter of the red inverse rod Q is Y, the fifth letter of inverse rod O is F, and Y and F are 6 apart on qwertzu (FGHJKPY). If then we had a set up of inverse rods including the pair QO we could use the series of numbers of line 8 of the mystic numbers to tell us on which sheets the various pairs should be looked up. However we can also use line 8 of this table on many occasions. Suppose for example that the pair ES of inverse rods is up. The series of sheets on which we had to look is again given by line 8, but we have to start in the third column under E instead of at the beginning under Q. Quite a convenient arrangement is to have the lines of the table written out on rods in gauge with the inverse rods and of double length. (This was once done for the service machine wheel III). Three lines of the table were put onto three sides of Mr Knox’s blank wooden inverse rods, and the fourth side occupied with the letters of the diagonal, in that case ABCD….. It was
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Back to Turing's Treatise on the Enigma. Chapter V.