General Report on Tunny
24W Page 133
characters are the same? The proportional bulge corresponding to odds
ζ0 is  |
Therefore, by the theorem of the chain of witnesses,
 |
and this gives the relation between π, θ, θ' in the slide-rule form.
(c) Wrong convergences of a rectangle and methods of starting.
A rigorous solution of the problem of the number of different crude convergences of a (1+2) rectangles seem to be very hard to find. However, two quite distinct attempts to solve the problem have been made. The first one (R1 pp 56, 57) tends to show that there are not more than 31 possible convergences*. The second one (R2 p 10, etc.) shows that for a rectangle of length 1271 probably at least 20 convergences are to be expected.
A very striking example of a wrong convergence occurred in about February, 1944. A message was converged twice on Colossus from two different random starts (R1 p 93, R2 p 14) and the same result was obtained each time. The rectangle was then converged by hand, using intelligence in the selection of a start and a very much better convergence was obtained (which was then checked on Colossus when the other wheels were broken). As an experiment, accurate convergence was applied to the original convergence and after a few steps it began to improve and become the same as the better convergence (R2 pp 21 etc.). Since that time more care was used in starting the convergence, but the accurate method was used only at first, and when there was not a flood of work.
One method of starting convergence is by the use of a skeleton (see e.g. R2 p 82, R4 p 20). This has the advantage that most of the arithmetic is avoided and a flag 16 by 16 can be made in the same time as a much smaller flag of the ordinary type. This method is not suitable for rectangles of length 2n x 1271(n = 2, 3, 4) and since so many of the rectangles were of length 8 x 1271 the method was not generally adopted (R3 p 74). The method is a special case of throwing away a lot of the smaller pieces of evidence in order to be able to work more quickly with the larger pieces (see R2 p 94).
* In fact there are exactly 31 convergences for 'scalar product' convergence.
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