\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}


\begin{document}

\module{The Enigma machine}{03}{16}{2 periods}

\prereq{Main cryptographic techniques}

\object{\begin{itemize}
	\item Discover the workings of the Enigma machine.
        \item Count the number of possibilities offered by the machine.
	\end{itemize}}

\mater{\begin{itemize}
	\item $6$ copies of each fact sheet.
	\item $6\times 3\times 3=54$ copies of the Enigma machine setup page.
	\item $6\times 4=24$ copies of the rotors page.
	\end{itemize}}

\modpart{Expert teams}{25 mins}

The class is divided in 6 groups. Each group is given a fact sheet about one of the aspects of the Enigma machine. They
have 25 minutes to understand the explanation and find a partial formula for the number of possibilities offered by
their part of the machine.


\modpart{Mixing the teams}{30 mins}

The class is once again divided in 6 groups, with one member from each of the 6 expert groups. They have 30 minutes to
communicate and understand the global workings of the Enigma machine, and compute the total number of possibilities.


\modpart{Coding and decoding messages}{1 period}

Each group has to code and decode some messages and gets mark depending on the number of correct codings and decodings.


\pagebreak

\moddocdis{The overall design of Enigma}{03}{16}{Fact sheet 1}

\begin{minipage}{10cm}
An Enigma machine is an electro-mechanical rotor machines that was used for the encryption and decryption
of secret messages by the German army during World War II. Like other rotor machines, the Enigma machine is a
combination of mechanical and electrical subsystems.

\bigskip

The mechanical subsystem consists of a keyboard; a set of rotating disks called rotors arranged adjacently along a
spindle; and one of various stepping components to turn one or more of the rotors with each key press.

\end{minipage}
\begin{minipage}{6cm}
\begin{center}
\includegraphics[width=5cm]{images/enigmainuse.eps}
\end{center}
\end{minipage}


\bigskip

The mechanical parts act in such a way as to form a varying electrical circuit. When a letter key is pressed, the
circuit is completed; current flows through the various components in their current configuration and ultimately lights
one of the display lamps, indicating the output letter. For example, when encrypting a message starting with
the letters ANX\ldots, the operator would first press the A key, and the Z lamp might light, so Z would be the first
letter of the ciphertext. The operator would next press N, and then X in the same fashion, and so on.

\bigskip



\begin{minipage}{8cm}
\begin{center}
\includegraphics[width=7cm]{images/enigmawiring.eps}
\end{center}
\end{minipage}
\begin{minipage}{8cm}
To illustrate the detailed operation of Enigma, please refer to the wiring diagram on the left. To simplify the example,
only four components of a complete Enigma machine are shown. In reality, there are 26 lamps and keys, several plugs
(varied with model) and rotor wirings inside the rotors (at least three were installed). 


Current flows from the battery {\bf (1)} through a depressed bi-directional letter-switch on a keyboard {\bf (2)} to the
plugboard {\bf (3)}. The current winds through the (unused in this instance, so shown closed) plug {\bf (3)}, then via
the entry wheel {\bf (4)} through the wiring of the three installed rotors {\bf (5)}, and enters the reflector {\bf
(6)}. The reflector returns the current, via an entirely different path, through the rotors {\bf (5)} and entry wheel
{\bf (4)}, proceeding through plug 'S' connected with a cable {\bf (8)} to plug 'D', and another bi-directional switch
{\bf (9)} to light the appropriate lamp.
\end{minipage}

\pagebreak



In German military usage, communications were divided up into a number of different networks, all using different
settings for their Enigma machines. Each unit operating on a network was assigned a settings list for its Enigma
for a period of time. For a message to be correctly encrypted and decrypted, both sender and receiver had to set up
their Enigma in the same way; the rotor selection and order, the starting position and the plugboard connections must be
identical. All these settings (together the key in modern terms) must have been established beforehand, and were
distributed in codebooks.

An Enigma machine's initial state, the cryptographic key, has several aspects:
\begin{itemize}
\item Wheel order (Walzenlage) : the choice of rotors and the order in which they are fitted.
\item Initial position of the rotors : chosen by the operator, different for each message.
\item  Ring settings (Ringstellung) : the position of the alphabet ring relative to the rotor wiring.
\item Plug settings (Steckerverbindungen) : the connections of the plugs in the plugboard.
\end{itemize}

\begin{center}
\includegraphics[width=15cm]{images/enigmacodebook.eps}
\end{center}

In fact, the Enigma cipher machine consists of five variable features: 
\begin{enumerate}
\item a plugboard which can contain from zero to thirteen dual-wired cables ;
\item three ordered (left to right) rotors which wire twenty-six input contact points to twenty-six output contact
points positioned on alternate faces of a disk ;
\item twenty-six serrations around the periphery of the rotors which allow the operator to specify an initial
rotational position for the rotors ;
\item a moveable ring on each of the rotors which controls the rotational behavior of the rotor immediately to the
left by means of a notch ;
\item a reflector half-rotor (which do not in fact rotate) to fold
inputs and outputs back onto the same face of contact points.
\end{enumerate}


{\bf\textsf{Your task :}} Understand the overall workings of the Enigma machine and be ready to put together the pieces
of information brought by other students. You will be the team leader.

 
\pagebreak

\moddocdis{The plugboard}{03}{16}{Fact sheet 2}

\begin{minipage}{11cm}
An Enigma machine is an electro-mechanical rotor machines that was used for the encryption and
decryption of secret messages by the German army during World War II. Like other rotor machines, the Enigma machine is a
combination of mechanical and electrical subsystems.

\bigskip

The first variable component was the plugboard. Twenty-six (A to Z) dual-holed sockets were on the front panel of
the machine. A dual-wired plugboard cable could be inserted making a connection between any pair of letters. Enigma
cryptographers had a choice of how many different cables could be inserted (from zero to thirteen)
and which letters were connected.
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\includegraphics[width=4cm]{images/enigma.eps}
\end{center}
\end{minipage}

\bigskip

A cable placed onto the plugboard connected letters up in pairs; for example, E and Q might be a connected pair. The
effect was to swap those letters before and after the main rotor scrambling unit. For example, when an operator presses
E, the signal was diverted to Q before entering the rotors. Several such steckered pairs, up to 13, might be used at one
time. However, normally only 10 pairs were used at any one time.

The plugboard contributed a great deal to the strength of the machine's encryption: more than an extra rotor would have
done. Enigma without a plugboard (known as unsteckered Enigma) can be solved relatively straightforwardly using hand
methods; these techniques are generally defeated by the addition of a plugboard, and Allied cryptanalysts resorted to
special machines to solve it.

\begin{center}
\includegraphics[width=10cm]{images/plugboard.eps}
\end{center}

{\bf\textsf{Your task :}} Understand the workings of the plugboard and find out the number of different settings there
were, depending on the number $p$ of plugs used. You will be the plugboard specialist in your team.


\pagebreak

\moddocdis{Rotors inner circuitry}{03}{16}{Fact sheet 3}

\begin{minipage}{11cm}
An Enigma machine is an electro-mechanical rotor machines that was used for the encryption and
decryption of secret messages by the German army during World War II. Like other rotor machines, the Enigma machine is a
combination of mechanical and electrical subsystems.


\bigskip

The second variable component was the three ordered (left to right) rotors which connected twenty-six input contact
points to twenty-six output contact points positioned on alternate faces of a disc.
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\includegraphics[width=4cm]{images/enigma.eps}
\end{center}
\end{minipage}

\bigskip

The rotors (alternatively wheels or drums, Walzen in German) formed the heart of an Enigma machine. Each rotor was a
disc approximately 10 cm (3.9 in) in diameter with brass spring-loaded pins on one
face arranged in a circle; on the other side are a corresponding number of circular electrical contacts. The pins and
contacts represent the alphabet -- typically the 26 letters A to Z. When the rotors were mounted side-by-side on the
spindle, the pins of one rotor rest against the contacts of the neighbouring rotor, forming an electrical connection.
Inside the body of the rotor, 26 wires connected each pin on one side to a contact on the other in a complex pattern.
Most of the rotors were identified by Roman numerals and each issued copy of rotor I was wired identically to all
other rotors I.

\bigskip

By itself, a rotor will perform only a very simple type of encryption -- a simple substitution cipher. For example, the
pin corresponding to the letter E might be wired to the contact for letter T on the opposite face, and so on. The
Enigma's complexity, and cryptographic security, came from using several rotors in series (usually three) and
the regular stepping movement of the rotors, thus implementing a poly-alphabetic substitution cipher.

\begin{center}
\includegraphics[height=5cm]{images/rotorstack.eps}\hspace*{1cm}
\includegraphics[height=5cm]{images/rotorsupanddown.eps}
\end{center}

{\bf\textsf{Your task :}} Understand the inner workings of the rotors and find out the number of different settings
there were for each disk, and for three consecutive disks. You will be the inner rotors specialist in your team.

\pagebreak

\moddocdis{Rotors positioning}{03}{16}{Fact sheet 4}

\begin{minipage}{11cm}
An Enigma machine is an electro-mechanical rotor machines that was used for the encryption and
decryption of secret messages by the German army during World War II. Like other rotor machines, the Enigma machine is a
combination of mechanical and electrical subsystems.


\bigskip

The third variable component of Enigma was the initial rotational position of the three rotors containing the wired
discs. This was specified by the cryptographers and set by the machine operators by means of twenty-six serrations
around the rotor periphery.

\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\includegraphics[width=4cm]{images/enigma.eps}
\end{center}
\end{minipage}

\bigskip

The rotors (alternatively wheels or drums, Walzen in German) formed the heart of an Enigma machine. By itself, a rotor
will perform only a very simple type of encryption -- a simple substitution cipher. For example, the pin corresponding
to the letter E might be wired to the contact for letter T on the opposite face, and so on. The Enigma's complexity, and
cryptographic security, came from using several rotors in series (usually three or four) and the regular stepping
movement of the rotors, thus implementing a poly-alphabetic substitution cipher. 

\bigskip

When placed in an Enigma, each rotor can be set to one of 26 possible positions. When inserted, it can be turned by hand
using the grooved finger-wheel which protrudes from the internal Enigma cover when closed. So that the operator can know
the rotor's position, each had an alphabet tyre (or letter ring) attached to the outside of the rotor disk, with 26
characters (typically letters); one of these could be seen through the window, thus indicating the rotational position
of the rotor.

\bigskip

The Army and Air Force Enigmas were used with several rotors. From December 1938, there were five, from which three
were chosen for insertion in the machine for a particular operating session. Rotors were marked with Roman numerals to
distinguish them: I, II, III, IV and V.

\begin{center}
\includegraphics[height=5cm]{images/rotorstack.eps}\hspace*{1cm}
\includegraphics[height=5cm]{images/rotorsbox.eps}
\end{center}

{\bf\textsf{Your task :}} Understand rotors positioning and find out the number of different settings
there were for each disk, and for three consecutive disks chosen among five possible disks. You will be the rotors
positioning specialist in your team.

\pagebreak

\moddocdis{Stepping motion}{03}{16}{Fact sheet 5}

\begin{minipage}{11cm}
An Enigma machine is an electro-mechanical rotor machines that was used for the encryption and
decryption of secret messages by the German army during World War II. Like other rotor machines, the Enigma machine is a
combination of mechanical and electrical subsystems.

\bigskip

The fourth variable component of the machine was a moveable ring on each of the rotors; each ring contained a notch in a
specific location. The purpose of the notch was to force a rotation of the rotor immediately to the left when the notch
was in a particular position

\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\includegraphics[width=4cm]{images/enigma.eps}
\end{center}
\end{minipage}

\bigskip


\begin{minipage}{8cm}
\begin{center}
\includegraphics[width=7.5cm]{images/rotorsmotion.eps}
\end{center}
\end{minipage}
\begin{minipage}{8cm}
The rotors (alternatively wheels or drums, Walzen in German) formed the heart of an Enigma machine. By itself, a rotor
will perform only a very simple type of encryption -- a simple substitution cipher. For example, the pin corresponding
to the letter E might be wired to the contact for letter T on the opposite face, and so on. 

\bigskip
To avoid merely implementing a simple (and easily breakable) substitution cipher, every key press caused one or more
rotors to step before the electrical connections were made, and so changed the substitution alphabet used for
encryption. This ensured that the cryptographic substitution would be different at each new rotor position, producing a
more formidable polyalphabetic substitution cipher.
\end{minipage}

\bigskip

\begin{minipage}{11cm}
The rightmost rotor rotated every time a key was pressed. The rightmost rotor's notch forced a rotation of the middle
rotor once every twenty-six keystrokes. The middle rotor's notch forced a rotation of the leftmost rotor once every 26 X
26 keystrokes. Since there were no more rotors, the leftmost rotor's notch had absolutely no effect whatsoever.
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\includegraphics[height=4cm]{images/rotors3D.eps}
\end{center}
\end{minipage}

{\bf\textsf{Your task :}} Understand the rotors stepping motion and find out the number of different settings
there were for the position of the notches for three consecutive disks. You will be the stepping motion
specialist in your team.

\pagebreak

\moddocdis{The reflector}{03}{16}{Fact sheet 6}

\begin{minipage}{11cm}
An Enigma machine is an electro-mechanical rotor machines that was used for the encryption and
decryption of secret messages by the German army during World War II. Like other rotor machines, the Enigma machine is a
combination of mechanical and electrical subsystems.

\bigskip

The fifth and final variable component of Enigma was the reflector. The reflector had twenty-six contact points like a
rotor, but only on one face.

\end{minipage}
\begin{minipage}{5cm}
\begin{center}
\includegraphics[width=4cm]{images/enigma.eps}
\end{center}
\end{minipage}


\bigskip

The reflector connected outputs of the last rotor in pairs, redirecting current back through the rotors by a different
route. The reflector ensured that Enigma is self-reciprocal: conveniently, encryption was the same as decryption.
However, the reflector also gave Enigma the property that no letter ever encrypted to itself. This was a severe
conceptual flaw and a cryptological mistake subsequently exploited by codebreakers.

\bigskip

Thirteen wires internally connected the twenty-six contact points together in a series of pairs so that a
connection coming in to the reflector from the rotors was sent back through the rotors a second time by a different
route. The internal wiring could be constructed in the following fashion. When one end of the first wire was connected
to contact point \#1, the other side of the wire had twenty-five different contact points to which it could be
connected.

\bigskip

Thus the first wire consumed two contact points and had twenty-five different possibilities. The second wire
also consumed two contact points, and had only twenty-three different connection possibilities remaining from the
unconsumed contact points. The third wire consumed two more contact points and had twenty-one possibilities for
connection.


\begin{center}
\includegraphics[height=5cm]{images/enigmareflector.eps}\hspace*{1cm}
\includegraphics[height=5.5cm]{images/enigmaoverview.eps}
\end{center}


{\bf\textsf{Your task :}} Understand the role and workings of the reflector and find out the number of different
settings there were. You will be the reflector specialist in your team.


\pagebreak

\moddocdis{Coding and decoding with Enigma}{03}{16}{Exercises}

For each of the following exercises, you will have to set up a paper Enigma machine with :
\begin{itemize}
\item a choice of up to 10 plugs, each one witching two letters ;
\item three rotors from the five available -- for practical purposes you will first have to cut them out and tape two
copies of each rotor side by side ;
\item the initial position for each rotor ;
\item the position of the notch for the first two rotors, that is the position which, we reached, will make the next
rotor move.
\end{itemize}

For the first two exercises, the setup will be given and you will have to decode a message. For the last two, you will
have to agree on a setup with another group, code a message, send it to the other group and decode their message. Each
exercise is worth 5 points.

We will agree that the first rotor starts moving to the left after the first letter is coded (or decoded), and that we
start back from the initial position at the beginning of a each new message. Also remember that the coding and decoding
processes are exactly the same. 

\sectionblack{Exercise 1 -- A simple message with a simple setup}

\begin{itemize}
\item Use no cables.
\item Pick disks I, II III, in that order.
\item Set the starting positions as 1, 1, 1.
\item Set the notches positions at 26, 26, 26.
\item Decode the message NZAZL HICGI NF.%ATTACK AT DAWN
\end{itemize}


\sectionblack{Exercise 2 -- A simple message with a more complicated setup}

\begin{itemize}
\item Look up the configuration for the day 31 on the overall design sheet.
\item Set up the Enigma machine accordingly.
\item Decode the message JMFLR PBDBT ZDJH%ENNEMY DEFEATED 
\end{itemize}

\sectionblack{Exercise 3 -- Coding and decoding a short message}

\begin{itemize}
\item Agree on a complete setup with another group, then code a short message, with less than 20 characters.
\item Send the ciphered message to the other group through the teacher.
\item Decipher the message received from the other group.
\end{itemize}

\sectionblack{Exercise 4 -- Coding and decoding a longer message}

\begin{itemize}
\item Agree on a complete setup with another group, then code a message with at least 60 characters.
\item Send the ciphered message to the other group through the teacher.
\item Decipher the message received from the other group.
\end{itemize}


\pagebreak

\moddocdis{Coding and decoding with Enigma}{03}{16}{Setup}

{\bf\textsf{Disks used :}} I \hspace{0.3cm}--\hspace{0.3cm} II \hspace{0.3cm}--\hspace{0.3cm} III
\hspace{0.3cm}--\hspace{0.3cm} IV \hspace{0.3cm}--\hspace{0.3cm} V

\smallskip

{\bf\textsf{Starting position and notch for disk 1 :}} \underline{\qquad} and  \underline{\qquad}

\smallskip

{\bf\textsf{Starting position and notch for disk 2 :}} \underline{\qquad} and  \underline{\qquad}

\smallskip

{\bf\textsf{Starting position and notch for disk 3 :}} \underline{\qquad} and  \underline{\qquad}

\begin{center}
\begin{tabular}{|cccccccccccccccccccccccccc|}
\hline
\rnode{A1}{\scriptsize a}& \rnode{B1}{\scriptsize b}& \rnode{C1}{\scriptsize c}& \rnode{D1}{\scriptsize c}&
\rnode{E1}{\scriptsize e}&
\rnode{F1}{\scriptsize f}& \rnode{G1}{\scriptsize g}& \rnode{H1}{\scriptsize h}& \rnode{I1}{\scriptsize i}&
\rnode{J1}{\scriptsize j}&
\rnode{K1}{\scriptsize k}& \rnode{L1}{\scriptsize l}& \rnode{M1}{\scriptsize m}& \rnode{N1}{\scriptsize n}&
\rnode{O1}{\scriptsize o}&
\rnode{P1}{\scriptsize p}& \rnode{Q1}{\scriptsize q}& \rnode{R1}{\scriptsize r}& \rnode{S1}{\scriptsize s}&
\rnode{T1}{\scriptsize t}&
\rnode{U1}{\scriptsize u}& \rnode{V1}{\scriptsize v}& \rnode{W1}{\scriptsize w}& \rnode{X1}{\scriptsize x}&
\rnode{Y1}{\scriptsize y}&
\rnode{Z1}{\scriptsize z}\\
\rnode{A2}{$\circ$}& \rnode{B2}{$\circ$}& \rnode{C2}{$\circ$}& \rnode{D2}{$\circ$}& \rnode{E2}{$\circ$}&
\rnode{F2}{$\circ$}& \rnode{G2}{$\circ$}& \rnode{H2}{$\circ$}& \rnode{I2}{$\circ$}& \rnode{J2}{$\circ$}&
\rnode{K2}{$\circ$}& \rnode{L2}{$\circ$}& \rnode{M2}{$\circ$}& \rnode{N2}{$\circ$}& \rnode{O2}{$\circ$}&
\rnode{P2}{$\circ$}& \rnode{Q2}{$\circ$}& \rnode{R2}{$\circ$}& \rnode{S2}{$\circ$}& \rnode{T2}{$\circ$}&
\rnode{U2}{$\circ$}& \rnode{V2}{$\circ$}& \rnode{W2}{$\circ$}& \rnode{X2}{$\circ$}& \rnode{Y2}{$\circ$}&
\rnode{Z2}{$\circ$}\\[50pt]
\rnode{A2}{$\circ$}& \rnode{B2}{$\circ$}& \rnode{C2}{$\circ$}& \rnode{D2}{$\circ$}& \rnode{E2}{$\circ$}&
\rnode{F2}{$\circ$}& \rnode{G2}{$\circ$}& \rnode{H2}{$\circ$}& \rnode{I2}{$\circ$}& \rnode{J2}{$\circ$}&
\rnode{K2}{$\circ$}& \rnode{L2}{$\circ$}& \rnode{M2}{$\circ$}& \rnode{N2}{$\circ$}& \rnode{O2}{$\circ$}&
\rnode{P2}{$\circ$}& \rnode{Q2}{$\circ$}& \rnode{R2}{$\circ$}& \rnode{S2}{$\circ$}& \rnode{T2}{$\circ$}&
\rnode{U2}{$\circ$}& \rnode{V2}{$\circ$}& \rnode{W2}{$\circ$}& \rnode{X2}{$\circ$}& \rnode{Y2}{$\circ$}&
\rnode{Z2}{$\circ$}\\
\hline
&&&&&&&&&&&&&&&&&&&&&&&&&\\[2.9cm]
\hline
&&&&&&&&&&&&&&&&&&&&&&&&&\\[2.9cm]
\hline
&&&&&&&&&&&&&&&&&&&&&&&&&\\[2.9cm]
\hline
\rnode{A1}{$\circ$}& \rnode{B1}{$\circ$}& \rnode{C1}{$\circ$}& \rnode{D1}{$\circ$}& \rnode{E1}{$\circ$}&
\rnode{F1}{$\circ$}& \rnode{G1}{$\circ$}& \rnode{H1}{$\circ$}& \rnode{I1}{$\circ$}& \rnode{J1}{$\circ$}&
\rnode{K1}{$\circ$}& \rnode{L1}{$\circ$}& \rnode{M1}{$\circ$}& \rnode{N1}{$\circ$}& \rnode{O1}{$\circ$}&
\rnode{P1}{$\circ$}& \rnode{Q1}{$\circ$}& \rnode{R1}{$\circ$}& \rnode{S1}{$\circ$}& \rnode{T1}{$\circ$}&
\rnode{U1}{$\circ$}& \rnode{V1}{$\circ$}& \rnode{W1}{$\circ$}& \rnode{X1}{$\circ$}& \rnode{Y1}{$\circ$}&
\rnode{Z1}{$\circ$}\\[-5pt]
\rnode{A2}{}& & & & &
& & & & &
& & & & &
& & & & &
& & & & \rnode{Y2}{}&
\\[-5pt]
&\rnode{B2}{} & & & &
& & & & &
& & & & &
& &  \rnode{R2}{}& &\rnode{T2}{}  &
& & & &&
\rnode{Z2}{}\\[-5pt]
& &\rnode{C2}{} & & &
& & & & &
& & & & &
& &  & & &
\rnode{U2}{}& & & &&
\\[-5pt]
& & & \rnode{D2}{}& &
& & \rnode{H2}{}& \rnode{I2}{} & &
& & & & &
\rnode{P2}{}& &  & & &
& \rnode{V2}{} & \rnode{W2}{} & &&
\\[-5pt]
& & & &\rnode{E2}{} &
& & & & & 
& & & & &
& \rnode{Q2}{}&  & & &
& & & &&
\\[-5pt]
& & & & &\rnode{F2}{}
& & & & &
& & & & &
& &  &\rnode{S2}{} & &
& & & &&
\\[-5pt]
& & & & &
& \rnode{G2}{} & & & & 
& \rnode{L2}{} & \rnode{M2}{} & & \rnode{O2}{} &
& &  & & &
& & & &&
\\[-5pt]
& & & & &
&  & & &\rnode{J2}{} & 
&  & & & &
& &  & & &
& & &\rnode{X2}{} &&
\\[-5pt]
& & & & &
&  & & & & \rnode{K2}{}
&  & &\rnode{N2}{} & &
& &  & & &
& & & &&
\\
\hline
\end{tabular}
\ncline{A1}{A2}\ncline{A2}{Y2}\ncline{Y2}{Y1}
\ncline[linestyle=dashed]{B1}{B2}\ncline[linestyle=dashed]{B2}{R2}\ncline[linestyle=dashed]{R2}{R1}
\ncline{C1}{C2}\ncline{C2}{U2}\ncline{U2}{U1}
\ncline[linestyle=dashed]{D1}{D2}\ncline[linestyle=dashed]{D2}{H2}\ncline[linestyle=dashed]{H2}{H1}
\ncline{E1}{E2}\ncline{E2}{Q2}\ncline{Q2}{Q1}
\ncline[linestyle=dashed]{F1}{F2}\ncline[linestyle=dashed]{F2}{S2}\ncline[linestyle=dashed]{S2}{S1}
\ncline{G1}{G2}\ncline{G2}{L2}\ncline{L2}{L1}
\ncline[linestyle=dashed]{I1}{I2}\ncline[linestyle=dashed]{I2}{P2}\ncline[linestyle=dashed]{P2}{P1}
\ncline{J1}{J2}\ncline{J2}{X2}\ncline{X2}{X1}
\ncline[linestyle=dashed]{K1}{K2}\ncline[linestyle=dashed]{K2}{N2}\ncline[linestyle=dashed]{N2}{N1}
\ncline{M1}{M2}\ncline{M2}{O2}\ncline{O2}{O1}
\ncline[linestyle=dashed]{T1}{T2}\ncline[linestyle=dashed]{T2}{Z2}\ncline[linestyle=dashed]{Z2}{Z1}
\ncline{V1}{V2}\ncline{V2}{W2}\ncline{W2}{W1}
\end{center}









\pagebreak

\moddoc{Rotors}

\begin{center}
{\bf\textsf{Rotor I}}

\bigskip

\begin{tabular}{|cccccccccccccccccccccccccc|}
\hline
\rnode{A1}{$\circ$}& \rnode{B1}{$\circ$}& \rnode{C1}{$\circ$}& \rnode{D1}{$\circ$}& \rnode{E1}{$\circ$}&
\rnode{F1}{$\circ$}& \rnode{G1}{$\circ$}& \rnode{H1}{$\circ$}& \rnode{I1}{$\circ$}& \rnode{J1}{$\circ$}&
\rnode{K1}{$\circ$}& \rnode{L1}{$\circ$}& \rnode{M1}{$\circ$}& \rnode{N1}{$\circ$}& \rnode{O1}{$\circ$}&
\rnode{P1}{$\circ$}& \rnode{Q1}{$\circ$}& \rnode{R1}{$\circ$}& \rnode{S1}{$\circ$}& \rnode{T1}{$\circ$}&
\rnode{U1}{$\circ$}& \rnode{V1}{$\circ$}& \rnode{W1}{$\circ$}& \rnode{X1}{$\circ$}& \rnode{Y1}{$\circ$}&
\rnode{Z1}{$\circ$}\\[70pt]
\rnode{A2}{$\circ$}& \rnode{B2}{$\circ$}& \rnode{C2}{$\circ$}& \rnode{D2}{$\circ$}& \rnode{E2}{$\circ$}&
\rnode{F2}{$\circ$}& \rnode{G2}{$\circ$}& \rnode{H2}{$\circ$}& \rnode{I2}{$\circ$}& \rnode{J2}{$\circ$}&
\rnode{K2}{$\circ$}& \rnode{L2}{$\circ$}& \rnode{M2}{$\circ$}& \rnode{N2}{$\circ$}& \rnode{O2}{$\circ$}&
\rnode{P2}{$\circ$}& \rnode{Q2}{$\circ$}& \rnode{R2}{$\circ$}& \rnode{S2}{$\circ$}& \rnode{T2}{$\circ$}&
\rnode{U2}{$\circ$}& \rnode{V2}{$\circ$}& \rnode{W2}{$\circ$}& \rnode{X2}{$\circ$}& \rnode{Y2}{$\circ$}&
\rnode{Z2}{$\circ$}\\
\hline
\end{tabular}
\ncline{A1}{E2} \ncline[linestyle=dashed]{B1}{K2} \ncline{C1}{M2} \ncline[linestyle=dashed]{D1}{F2} \ncline{E1}{L2}
\ncline[linestyle=dashed]{F1}{G2} \ncline{G1}{D2} \ncline[linestyle=dashed]{H1}{Q2} \ncline{I1}{V2}
\ncline[linestyle=dashed]{J1}{Z2}
\ncline{K1}{N2} \ncline[linestyle=dashed]{L1}{T2} \ncline{M1}{O2} \ncline[linestyle=dashed]{N1}{W2} \ncline{O1}{Y2}
\ncline[linestyle=dashed]{P1}{H2} \ncline{Q1}{X2} \ncline[linestyle=dashed]{R1}{U2} \ncline{S1}{S2}
\ncline[linestyle=dashed]{T1}{P2}
\ncline{U1}{A2} \ncline[linestyle=dashed]{V1}{I2} \ncline{W1}{B2} \ncline[linestyle=dashed]{X1}{R2} \ncline{Y1}{C2}
\ncline[linestyle=dashed]{Z1}{J2}
\end{center}

\begin{center}
{\bf\textsf{Rotor II}}

\bigskip

\begin{tabular}{|cccccccccccccccccccccccccc|}
\hline
\rnode{A1}{$\circ$}& \rnode{B1}{$\circ$}& \rnode{C1}{$\circ$}& \rnode{D1}{$\circ$}& \rnode{E1}{$\circ$}&
\rnode{F1}{$\circ$}& \rnode{G1}{$\circ$}& \rnode{H1}{$\circ$}& \rnode{I1}{$\circ$}& \rnode{J1}{$\circ$}&
\rnode{K1}{$\circ$}& \rnode{L1}{$\circ$}& \rnode{M1}{$\circ$}& \rnode{N1}{$\circ$}& \rnode{O1}{$\circ$}&
\rnode{P1}{$\circ$}& \rnode{Q1}{$\circ$}& \rnode{R1}{$\circ$}& \rnode{S1}{$\circ$}& \rnode{T1}{$\circ$}&
\rnode{U1}{$\circ$}& \rnode{V1}{$\circ$}& \rnode{W1}{$\circ$}& \rnode{X1}{$\circ$}& \rnode{Y1}{$\circ$}&
\rnode{Z1}{$\circ$}\\[70pt]
\rnode{A2}{$\circ$}& \rnode{B2}{$\circ$}& \rnode{C2}{$\circ$}& \rnode{D2}{$\circ$}& \rnode{E2}{$\circ$}&
\rnode{F2}{$\circ$}& \rnode{G2}{$\circ$}& \rnode{H2}{$\circ$}& \rnode{I2}{$\circ$}& \rnode{J2}{$\circ$}&
\rnode{K2}{$\circ$}& \rnode{L2}{$\circ$}& \rnode{M2}{$\circ$}& \rnode{N2}{$\circ$}& \rnode{O2}{$\circ$}&
\rnode{P2}{$\circ$}& \rnode{Q2}{$\circ$}& \rnode{R2}{$\circ$}& \rnode{S2}{$\circ$}& \rnode{T2}{$\circ$}&
\rnode{U2}{$\circ$}& \rnode{V2}{$\circ$}& \rnode{W2}{$\circ$}& \rnode{X2}{$\circ$}& \rnode{Y2}{$\circ$}&
\rnode{Z2}{$\circ$}\\
\hline
\end{tabular}
\ncline{A1}{A2} \ncline[linestyle=dashed]{B1}{J2} \ncline{C1}{D2} \ncline[linestyle=dashed]{D1}{K2} \ncline{E1}{S2}
\ncline[linestyle=dashed]{F1}{I2} \ncline{G1}{R2} \ncline[linestyle=dashed]{H1}{U2} \ncline{I1}{X2}
\ncline[linestyle=dashed]{J1}{B2}
\ncline{K1}{L2} \ncline[linestyle=dashed]{L1}{H2} \ncline{M1}{W2} \ncline[linestyle=dashed]{N1}{T2} \ncline{O1}{M2}
\ncline[linestyle=dashed]{P1}{C2} \ncline{Q1}{Q2} \ncline[linestyle=dashed]{R1}{G2} \ncline{S1}{Z2}
\ncline[linestyle=dashed]{T1}{N2}
\ncline{U1}{P2} \ncline[linestyle=dashed]{V1}{Y2} \ncline{W1}{F2} \ncline[linestyle=dashed]{X1}{V2} \ncline{Y1}{O2}
\ncline[linestyle=dashed]{Z1}{E2}
\end{center}

\begin{center}
{\bf\textsf{Rotor III}}

\bigskip

\begin{tabular}{|cccccccccccccccccccccccccc|}
\hline
\rnode{A1}{$\circ$}& \rnode{B1}{$\circ$}& \rnode{C1}{$\circ$}& \rnode{D1}{$\circ$}& \rnode{E1}{$\circ$}&
\rnode{F1}{$\circ$}& \rnode{G1}{$\circ$}& \rnode{H1}{$\circ$}& \rnode{I1}{$\circ$}& \rnode{J1}{$\circ$}&
\rnode{K1}{$\circ$}& \rnode{L1}{$\circ$}& \rnode{M1}{$\circ$}& \rnode{N1}{$\circ$}& \rnode{O1}{$\circ$}&
\rnode{P1}{$\circ$}& \rnode{Q1}{$\circ$}& \rnode{R1}{$\circ$}& \rnode{S1}{$\circ$}& \rnode{T1}{$\circ$}&
\rnode{U1}{$\circ$}& \rnode{V1}{$\circ$}& \rnode{W1}{$\circ$}& \rnode{X1}{$\circ$}& \rnode{Y1}{$\circ$}&
\rnode{Z1}{$\circ$}\\[70pt]
\rnode{A2}{$\circ$}& \rnode{B2}{$\circ$}& \rnode{C2}{$\circ$}& \rnode{D2}{$\circ$}& \rnode{E2}{$\circ$}&
\rnode{F2}{$\circ$}& \rnode{G2}{$\circ$}& \rnode{H2}{$\circ$}& \rnode{I2}{$\circ$}& \rnode{J2}{$\circ$}&
\rnode{K2}{$\circ$}& \rnode{L2}{$\circ$}& \rnode{M2}{$\circ$}& \rnode{N2}{$\circ$}& \rnode{O2}{$\circ$}&
\rnode{P2}{$\circ$}& \rnode{Q2}{$\circ$}& \rnode{R2}{$\circ$}& \rnode{S2}{$\circ$}& \rnode{T2}{$\circ$}&
\rnode{U2}{$\circ$}& \rnode{V2}{$\circ$}& \rnode{W2}{$\circ$}& \rnode{X2}{$\circ$}& \rnode{Y2}{$\circ$}&
\rnode{Z2}{$\circ$}\\
\hline
\end{tabular}
\ncline{A1}{B2} \ncline[linestyle=dashed]{B1}{D2} \ncline{C1}{F2} \ncline[linestyle=dashed]{D1}{H2} \ncline{E1}{J2}
\ncline[linestyle=dashed]{F1}{L2} \ncline{G1}{C2} \ncline[linestyle=dashed]{H1}{P2} \ncline{I1}{R2}
\ncline[linestyle=dashed]{J1}{T2}
\ncline{K1}{X2} \ncline[linestyle=dashed]{L1}{V2} \ncline{M1}{Z2} \ncline[linestyle=dashed]{N1}{N2} \ncline{O1}{Y2}
\ncline[linestyle=dashed]{P1}{E2} \ncline{Q1}{I2} \ncline[linestyle=dashed]{R1}{W2} \ncline{S1}{G2}
\ncline[linestyle=dashed]{T1}{A2}
\ncline{U1}{K2} \ncline[linestyle=dashed]{V1}{M2} \ncline{W1}{U2} \ncline[linestyle=dashed]{X1}{S2} \ncline{Y1}{Q2}
\ncline[linestyle=dashed]{Z1}{O2}
\end{center}

\begin{center}
{\bf\textsf{Rotor IV}}

\bigskip

\begin{tabular}{|cccccccccccccccccccccccccc|}
\hline
\rnode{A1}{$\circ$}& \rnode{B1}{$\circ$}& \rnode{C1}{$\circ$}& \rnode{D1}{$\circ$}& \rnode{E1}{$\circ$}&
\rnode{F1}{$\circ$}& \rnode{G1}{$\circ$}& \rnode{H1}{$\circ$}& \rnode{I1}{$\circ$}& \rnode{J1}{$\circ$}&
\rnode{K1}{$\circ$}& \rnode{L1}{$\circ$}& \rnode{M1}{$\circ$}& \rnode{N1}{$\circ$}& \rnode{O1}{$\circ$}&
\rnode{P1}{$\circ$}& \rnode{Q1}{$\circ$}& \rnode{R1}{$\circ$}& \rnode{S1}{$\circ$}& \rnode{T1}{$\circ$}&
\rnode{U1}{$\circ$}& \rnode{V1}{$\circ$}& \rnode{W1}{$\circ$}& \rnode{X1}{$\circ$}& \rnode{Y1}{$\circ$}&
\rnode{Z1}{$\circ$}\\[70pt]
\rnode{A2}{$\circ$}& \rnode{B2}{$\circ$}& \rnode{C2}{$\circ$}& \rnode{D2}{$\circ$}& \rnode{E2}{$\circ$}&
\rnode{F2}{$\circ$}& \rnode{G2}{$\circ$}& \rnode{H2}{$\circ$}& \rnode{I2}{$\circ$}& \rnode{J2}{$\circ$}&
\rnode{K2}{$\circ$}& \rnode{L2}{$\circ$}& \rnode{M2}{$\circ$}& \rnode{N2}{$\circ$}& \rnode{O2}{$\circ$}&
\rnode{P2}{$\circ$}& \rnode{Q2}{$\circ$}& \rnode{R2}{$\circ$}& \rnode{S2}{$\circ$}& \rnode{T2}{$\circ$}&
\rnode{U2}{$\circ$}& \rnode{V2}{$\circ$}& \rnode{W2}{$\circ$}& \rnode{X2}{$\circ$}& \rnode{Y2}{$\circ$}&
\rnode{Z2}{$\circ$}\\
\hline
\end{tabular}
\ncline{A1}{E2} \ncline[linestyle=dashed]{B1}{S2} \ncline{C1}{O2} \ncline[linestyle=dashed]{D1}{V2} \ncline{E1}{P2}
\ncline[linestyle=dashed]{F1}{Z2} \ncline{G1}{J2} \ncline[linestyle=dashed]{H1}{A2} \ncline{I1}{Y2}
\ncline[linestyle=dashed]{J1}{Q2}
\ncline{K1}{U2} \ncline[linestyle=dashed]{L1}{I2} \ncline{M1}{R2} \ncline[linestyle=dashed]{N1}{H2} \ncline{O1}{X2}
\ncline[linestyle=dashed]{P1}{L2} \ncline{Q1}{N2} \ncline[linestyle=dashed]{R1}{F2} \ncline{S1}{T2}
\ncline[linestyle=dashed]{T1}{G2}
\ncline{U1}{K2} \ncline[linestyle=dashed]{V1}{D2} \ncline{W1}{C2} \ncline[linestyle=dashed]{X1}{M2} \ncline{Y1}{W2}
\ncline[linestyle=dashed]{Z1}{B2}
\end{center}

\begin{center}
{\bf\textsf{Rotor V}}

\bigskip

\begin{tabular}{|cccccccccccccccccccccccccc|}
\hline
\rnode{A1}{$\circ$}& \rnode{B1}{$\circ$}& \rnode{C1}{$\circ$}& \rnode{D1}{$\circ$}& \rnode{E1}{$\circ$}&
\rnode{F1}{$\circ$}& \rnode{G1}{$\circ$}& \rnode{H1}{$\circ$}& \rnode{I1}{$\circ$}& \rnode{J1}{$\circ$}&
\rnode{K1}{$\circ$}& \rnode{L1}{$\circ$}& \rnode{M1}{$\circ$}& \rnode{N1}{$\circ$}& \rnode{O1}{$\circ$}&
\rnode{P1}{$\circ$}& \rnode{Q1}{$\circ$}& \rnode{R1}{$\circ$}& \rnode{S1}{$\circ$}& \rnode{T1}{$\circ$}&
\rnode{U1}{$\circ$}& \rnode{V1}{$\circ$}& \rnode{W1}{$\circ$}& \rnode{X1}{$\circ$}& \rnode{Y1}{$\circ$}&
\rnode{Z1}{$\circ$}\\[70pt]
\rnode{A2}{$\circ$}& \rnode{B2}{$\circ$}& \rnode{C2}{$\circ$}& \rnode{D2}{$\circ$}& \rnode{E2}{$\circ$}&
\rnode{F2}{$\circ$}& \rnode{G2}{$\circ$}& \rnode{H2}{$\circ$}& \rnode{I2}{$\circ$}& \rnode{J2}{$\circ$}&
\rnode{K2}{$\circ$}& \rnode{L2}{$\circ$}& \rnode{M2}{$\circ$}& \rnode{N2}{$\circ$}& \rnode{O2}{$\circ$}&
\rnode{P2}{$\circ$}& \rnode{Q2}{$\circ$}& \rnode{R2}{$\circ$}& \rnode{S2}{$\circ$}& \rnode{T2}{$\circ$}&
\rnode{U2}{$\circ$}& \rnode{V2}{$\circ$}& \rnode{W2}{$\circ$}& \rnode{X2}{$\circ$}& \rnode{Y2}{$\circ$}&
\rnode{Z2}{$\circ$}\\
\hline
\end{tabular}
\ncline{A1}{V2} \ncline[linestyle=dashed]{B1}{Z2} \ncline{C1}{B2} \ncline[linestyle=dashed]{D1}{R2} \ncline{E1}{G2}
\ncline[linestyle=dashed]{F1}{I2} \ncline{G1}{T2} \ncline[linestyle=dashed]{H1}{Y2} \ncline{I1}{U2}
\ncline[linestyle=dashed]{J1}{P2}
\ncline{K1}{S2} \ncline[linestyle=dashed]{L1}{D2} \ncline{M1}{N2} \ncline[linestyle=dashed]{N1}{H2} \ncline{O1}{L2}
\ncline[linestyle=dashed]{P1}{X2} \ncline{Q1}{A2} \ncline[linestyle=dashed]{R1}{W2} \ncline{S1}{M2}
\ncline[linestyle=dashed]{T1}{J2}
\ncline{U1}{Q2} \ncline[linestyle=dashed]{V1}{O2} \ncline{W1}{F2} \ncline[linestyle=dashed]{X1}{E2} \ncline{Y1}{C2}
\ncline[linestyle=dashed]{Z1}{K2}
\end{center}

\end{document}
