# General Report on Tunny

11C Page 11

11C     WHEEL PATTERNS.

(a) German Precautions.

Though the Germans never fully appreciated the weaknesses of the Tunny machine, they were alive to some of the more elementary pitfalls. In particular they took care to construct wheel patterns so that
(1) there were not too many extensions of the psis.
(2) there was an equal number of dots and crosses in each impulse of the chi-stream and the extended psi-stream.
(3) the sum of consecutive characters in each impulse of the chi-stream and the extended psi-stream was dot and cross with equal frequency.

(b) Differenced and Undifferenced Wheels.

The letter (or character) obtained by adding any letter (or character) to its successor is known as the differenced or delta (Δ) letter (or character) e.g.

 ΔP = P + P ΔΧ = Χ + Χ ΔΨ3' = Ψ3' + Ψ3'

A differenced wheel pattern is obtained by adding each character in a wheel pattern to its successor, and will clearly have the same period as the undifferenced pattern:

 Χ4 pattern : . . x x . x . x x x . . x . . x x . . . x . . x x x ΔΧ4 pattern : . x . x x x x . . x . x x . x . x . . x x . x . . x

It will be readily seen that the number of crosses in the differenced wheel pattern is equal to twice the number of 'groups' of crosses (or dots) in the undifferenced pattern and is therefore even.

(c) Construction of Wheel Patterns.

Conditions (2) and (3) above were fulfilled as far as the chi-stream was concerned by the rule that the number of crosses in each Χ and ΔΧ pattern should be (as nearly as possible) half the length of the wheel.

The number of crosses in all undifferenced Ψ patterns was also made (as nearly as possible) half the length of the wheel. When the psi was extended, the extension produced additional dots or crosses and the proportion was preserved.

The case of the differenced Ψ patterns is different. At every extension, a letter of the Ψ' stream is repeated and therefore there is a stroke in the ΔΨ stream. A dot is therefore added to each impulse of the ΔΨ' stream at each extension, and therefore, in order to preserve an equal number of dots and crosses in each impulse of the ΔΨ' stream (after extension) there must be a preponderance of crosses in each ΔΨ pattern (before extension).

(d) The Law ab = ½.

The proportion of crosses in the TM stream is called a.
The proportion of crosses in each ΔΨ pattern is called b.

The Germans wished to ensure that the proportion of dots and crosses in each impulse of the ΔΨ' stream was (if possible) equal to ½.

Now, at TM dots, there is an extension in the ΔΨ' and therefore a stroke. So a cross in any impulse of ΔΨ' must occur against a TM cross.

In each impulse, TM crosses occur a proportion a of the time, and at a proportion b of TM cross positions there is a ΔΨ cross. Therefore proportion of crosses in each impulse of ΔΨ' stream = ab.