askets as if they were Nip pillboxes, and
Lieutenant Ethridge seems mollified. Bobby Shaftoe, having scored massive
points, leads Privates Daniels, Nathan, and Branph out into the street
before Lieutenant Ethridge figures out that he was just guessing. They head
for the meat locker up on the ridge, double time.
These Marines are all lethal combat veterans or else they never would
have gotten into a mess this bad trapped on a gratuitously dangerous
continent (Africa) surrounded by the enemy (United States Army troops).
Still, when they get into that locker and take their first gander at PFC
Hott, a hush comes over them.
Private Branph clasps his hands, rubbing them together surreptitiously.
"Dear Lord "
"Shut up, Private!" Shaftoe says, "I already did that."
"Okay, Sarge."
"Go find a meat saw!" Shaftoe says to Private Nathan.
The privates all gasp.
"For the fucking pig!" Shaftoe clarifies. Then he turns to Private
Daniels, who is carrying a featureless bundle, and says, "Open it up!"
The bundle (which was issued by Ethridge to Shaftoe) turns out to
contain a black wetsuit. Nothing GI; some kind of European model. Shaftoe
unfolds it and examines its various parts while Privates Nathan and Branph
dismember Frosty the Pig with vigorous strokes of an enormous bucksaw.
They are all working away silently when a new voice interrupts. "Dear
Lord," the voice begins, as they all look up to see a man standing nearby,
hands clasped prayerfully. His words, sacramentally condensed into an
outward and visible cloud of steam, veil his face. His uniform and rank are
obscured by an Army blanket thrown over his shoulders. He'd look like a
camel riding Holy Land prophet if he were not clean shaven and wearing Rape
Prevention Glasses.
"Goddamn it!" Shaftoe says. "I already said a fucking prayer."
"But are we praying for Private Hott, or for ourselves?" the man says.
This is a poser. Everything becomes quiet as the meat saw stops moving.
Shaftoe drops the wetsuit and stands up. Blanket Man's got very short
grizzly hair, or maybe that's frost coalescing on his scalp. His ice colored
eyes meet Shaftoe's through the mile thick lenses of his RPGs, as if he's
really expecting an answer. Shaftoe takes a step closer and realizes that
the man is wearing a clerical collar.
"You tell me, Rev," Shaftoe says.
Then he recognizes Blanket Man. He's about to let fly with a lusty What
in the fuck are you doing here, but something makes him hold back. The
chaplain's eyes make a sideways dart so small and so fast that only Shaftoe,
who's practically rubbing noses with him, could possibly see it.
The message being: Shut up, Bobby, we'll talk later.
"Private Hott is with God now or wherever people go after they die,"
says Enoch "You can call me Brother" Root.
"What kind of an attitude is that!? Course he's with God. Jesus Christ!
'Wherever they go when they die.' What kind of a chaplain are you?"
"I guess I'm a Detachment 2702 kind of chaplain," the chaplain says.
Lieutenant Enoch Root finally breaks eye contact with Shaftoe and turns his
gaze to where the action is. "As you were, fellows," he says. "Looks like
bacon tonight, huh?"
The men chuckle nervously and resume sawing.
Once they get the pig's carcass disentangled from Hott's, each of the
Marines grabs a limb. They carry Hott out into the butcher shop, which has
been temporarily evacuated for purposes of this operation, so that Hott's
former comrades in shanks will not spread rumors.
Hasty evacuation of a butcher shop after one of its workers has been
found dead on the floor could spawn a few rumors in and of itself. So the
cover story du jour, freshly spun by Lieutenant Ethridge, is that Detachment
2702 is (contrary to all outward appearances) an elite, crack medical team
concerned that Hott had been struck down by a rare new form of North African
food poisoning. Maybe even something deliberately left behind by the French,
who are, by accounts, a little irritable about having their battleship sunk.
Anyway, the whole shop (the story goes) has to be shut down for the day and
gone over with a nit comb. Hott's corpse will be cremated before being sent
back to the family, just to make sure that the dreaded affliction does not
spread into Chicago the planetary abbatoir capital where its incalculable
consequences could alter the outcome of the war.
There is a GI coffin laid out on the floor, just to preserve the
fiction. Shaftoe and his men ignore it completely and begin dressing the
body, first in an appalling pair of swim trunks, then various components of
the wetsuit.
"Hey!" Ethridge says. "I thought you were going to do the gloves last."
"Sir, we're doing them first, by your leave, sir!" Bobby Shaftoe says.
"On account of his fingers will thaw out first and once that happens we
are screwed, sir!"
"Well, slap this on him first," Ethridge says, and hands over a wrist
watch. Shaftoe hefts it and whistles. It's a beaut: a Swiss chronometer in
solid uranium, its jewel laden movement throbbing away like the heart beat
of a small mammal. He swings it on the end of its wristband, made in
cunningly joined armor plates. It is heavy enough to stun a muskellunge.
"Nice," Shaftoe says, "but it doesn't tell time too good."
"In the time zone where we are going," Ethridge says, "it does."
The chastened Shaftoe sets about his work. Meanwhile, Lieutenants
Ethridge and Root are making themselves useful. They carry the crudely sawed
remains of Frosty the Pig into the butcher shop and throw them on a gigantic
scale. They add up to some thirty kilograms, whatever the fuck that means.
Enoch Root, showing an appetite for physical labor that is duly and silently
noted by the men, hauls in another pig carcass, stiff as a Radio Flyer, and
dumps it onto the scale, bringing the total up to seventy. Ethridge does the
breaststroke through clouds of flies to gather up all the cuts of meat that
were on the chopping blocks when the place was evacuated. He throws them on
the scale and the needle swings up to near the one hundred mark. From that
point they are able to bring it up to one thirty by ferrying hams and roasts
in from the freezer one at a time. Enoch Root who seems to be conversant
with exotic systems of measurement has made a calculation, and checked it
twice, establishing that the weight of Gerald Hott, converted into
kilograms, is one hundred and thirty.
All the meat goes into the coffin. Ethridge slams the lid shut,
trapping some flies who have no idea what they are in for. Root goes around
with a clawhammer, driving in sixteen penny nails with sure, powerful,
Carpenter of Nazareth like strokes. Meanwhile, Ethridge has taken a GI
manual out of his briefcase. Shaftoe is close enough to read the title,
printed in block letters on its olive drab cover:
COFFIN SEALING PROCEDURES
PART III: TROPICAL ENVIRONMENTS
VOL. II: HIGH DISEASE RISK SITUATIONS (BUBONIC PLAGUE, ETC.)
The two lieutenants devote a good hour to following the instructions in
that manual. The instructions are not that complicated, but Enoch Root keeps
noticing syntactical ambiguities and wants to explore their ramifications.
First this rattles Ethridge, then his emotions tend towards impatience and,
finally, extreme pragmatism. To make the chaplain shut up, Ethridge
confiscates the manual and starts Root on stenciling Hott's name on the
coffin and pasting it up with red stickers printed with medical warnings so
appalling that the topic headings alone induce faint nausea. By the time
Root is finished, the only person who can legally open this coffin is
General George C. Marshall himself, and even he would have to first get
special permission from the Surgeon General and evacuate all living things
within a hundred mile radius.
"Chaplain talks kind of funny," says Private Nathan at one point,
listening, slackjawed, to one of these Root/Ethridge debates.
"Yeah!" exclaims Private Branph, as if the accent took a really keen
listener to notice. "What kind of an accent is that anyway?"
All eyes turn to Bobby Shaftoe, who pretends to listen for a bit and
then says, "Well, fellas, I would guess that this Enoch Root is the
offspring of a long line of Dutch and possibly German missionaries in the
South Sea Islands, interbred with Aussies. And furthermore, I would guess
that being as how he grew up in territories controlled by the British that
he carries a British passport and was drafted into their military when the
war started and is now part of ANZAC."
"Haw!" roars Private Daniels, "if you got all of that right, I'll give
you five bucks ."
"Deal," Shaftoe says.
Ethridge and Root finish sealing the coffin at about the same time
Shaftoe and his Marines are wrestling the last bits of the wetsuit into
place. It takes a shitload of talcum powder, but they get it done. Ethridge
supplies them with the talcum powder, which is not GI talc; it is from
somewhere in Europe. Some of the letters on the label have pairs of dots
over them, which Shaftoe knows to be a characteristic of the German
language.
A truck backs up to the loading dock, smelling the fresh paint (it is a
Detachment 2702 truck). In go the sealed coffin and the now vulcanized dead
butcher.
"I'm going to stay behind and check the wastebaskets," Lieutenant
Ethridge tells Shaftoe. "I'll meet you at the airfield in one hour."
Shaftoe imagines one hour in the back of a hot truck with this cargo.
"You want me to keep him on ice, sir?" he asks.
Ethridge has to think about this one for a while. He sucks his teeth,
checks his watch, hems and haws. But when he finally answers, he sounds
definite. "Negative. It is imperative, for purposes of this mission, that we
now get him into a thawed mode."
PFC General Hott and his meat laden coffin occupy the center of the
truck's bed. The Marines sit to the sides, arranged like pallbearers.
Shaftoe finds himself staring across the carnage into the face of Enoch
Root, which is wearing an expression of forced nonchalance.
Shaftoe knows he ought to wait, but he just can't stand it. "What are
you doing here?" he finally says.
"The detachment is relocating," the Rev says. "Closer to the front."
"We just got off the fucking boat," Shaftoe says. "Of course we're
going closer to the goddamn front we can't go any farther unless we swim ."
"As long as we're pulling up stakes," Root says coolly, "I'll be coming
along for the ride."
"I don't mean that," Bobby Shaftoe says. "I mean, why should the
detachment have a chaplain?"
"You know the military," Root says. "Every unit has to have one."
"It's bad luck."
"It's bad luck to have a chaplain? Why?"
"It means the waffle butts are expecting a lot of funerals, is why."
"So you are taking the position that the only thing a cleric can do is
to preside over funerals? Interesting."
"And weddings and baptisms," Shaftoe says. All of the other Marines
chortle.
"Could it be you're feeling a little anxious about the unusual nature
of Detachment 2702's first mission?" Root inquires, casting a significant
glance at the late Hott, then staring directly into Shaftoe's eyes.
"Anxious? Listen, Rev, I done some things on Guadalcanal that make this
look like Emily Fucking Post."
All of the other Marines think this is a great line, but Root is
undeterred.
"Did you know why you were doing those things on Guadalcanal?"
"Sure! To stay alive."
"Do you know why you're doing this?"
"Fuck no."
"Doesn't that irritate you a little bit? Or are you too much of a
stupid jarhead to care?"
"Well, you kind of backed me into a corner there, Rev," Shaftoe says.
After a pause he goes on, "I'll admit to being a little curious.
"If there were someone in Detachment 2702 who could help answer your
questions about why, would that be useful?"
"I guess so," Shaftoe grumbles. "It just seems weird to have a
chaplain."
"Why does it seem weird?"
"Because of what kind of unit this is."
"What kind of unit is it?" Root asks. He asks it with a certain
sadistic pleasure.
"We're not supposed to talk about it," Shaftoe says. "And anyway, we
don't know."
Down the hill, immense zigzagging ramps descend pompously over rows of
tiger striped arches to the strand of ramifying railway lines that feed the
port from the south. "It's like standing in the drain of a fucking pinball
machine," says B. Shaftoe, looking up at the way they have just come,
thinking about what might come rolling down out of the Casbah. They head
south along those railway lines and come into a zone of ore dumps and coal
heaps and smokestacks, clearly recognizable to Great Lakes Eagle Scout
Shaftoe, but here operated through some kind of cross cultured gear train
about a million meshings deep. They pull up in front of the Société
Algérienne d'Éclairage et de Force, a double smokestacked behemoth with the
biggest coal pile of all. They're in the middle of nowhere, but it's obvious
that they are expected. Here as everywhere else that Detachment 2702 goes a
strange Rank Inflation Effect is taking place. The coffin is carried into
the SAEF by two lieutenants, a captain, and a major, overseen by a colonel!
There is not a single enlisted man in sight, and Bobby Shaftoe, a mere
sergeant, worries about what sort of work they'll find for him. There is
also a Paperwork Negation Effect going on here; whenever Shaftoe expects to
be stalled by the usual half an hour's worth of red tape, an anxious officer
runs up and waves his hands furiously and he is allowed to proceed.
An Arab, wearing what appears to be a red coffee can on his head, hauls
an iron door open; flames lunge at him and he beats them back with a
blackened iron stick. The pallbearers center the head of the coffin in the
opening and then shove it through, like ramming a big shell home into a
sixteen inch gun, and the man with the can on his head clangs the door shut,
a tassel on the top of his can whipping around crazily. Before he's even got
it latched he's yodeling just like those guys up in the Casbah. The officers
all stand around agreeing with each other and signing their names on
clipboards.
So with a dearth of complications that can only strike combat veteran
Bobby Shaftoe as eerie, the truck leaves the Société Algérienne d'Éclairage
et de Force behind and heads back up those damn ramps into Algiers. The
climb's steep a first gear project all the way. Vendors with push carts
loaded with boiling oil are not only keeping up with them but cooking
fritters along the way. Three legged dogs run and fight underneath the
actual drive train of the truck. Detachment 2702 is also dogged by coffee
can wearing natives threatening to play guitars made of jerry cans, and by
orange vendors and snake charmers, and a few blue eyed burnoose wearers
holding up lumps of unwrapped and unlabelled dark stuff. Like hailstones,
these may be classified by analogy to fruits and sporting goods. Typically
they range from grape to baseball. At one point, the chaplain impulsively
trades a Hershey bar for a golf ball of the stuff.
"What is that? Chocolate?" Bobby Shaftoe asks.
"If it was chocolate," Root says, "that guy wouldn't have taken a
Hershey bar for it."
Shaftoe shrugs. "Unless it's shitty chocolate."
"Or shit!" blurts Private Nathan, provoking incredible hilarity.
"You heard of Mary Jane?" Root asks.
Shaftoe role model, leader of men stifles the impulse to say, Heard of
her? I've fucked her!
"This is the concentrated essence," says Enoch Root.
"How would you know, Rev?" says Private Daniels.
The Rev is not rattled. "I'm the God guy here, right? I know the
religious angle?"
"Yes, sir!"
"Well, at one time, there was a group of Muslims called the hashishin
who would eat this stuff and then go out and kill people. They were so good
at it, they became famous or infamous. Over time the pronunciation of the
name has changed we know them as assassins."
There is an appropriately respectful silence. Finally, Sergeant Shaftoe
says, "What the hell are we waiting for?"
They eat some. Shaftoe, being the highest ranking enlisted man present,
eats more than the others. Nothing happens. "Only person I feel like
assassinating is that guy who sold it to us," he says.
***
The airfield, eleven miles out of town, is busier than it was ever
intended to be. This is nice grape– and olive growing land, but stony
mountains are visible farther inland, and beyond 'em is a patch of sand the
size of the United States most of which seems to be airborne and headed
their way. Countless airplanes predominantly Dakota transports, a.k.a.
Gooney Birds stir up vast, tongue coating, booger nucleating dust clouds. It
doesn't occur to Shaftoe for quite some time that his dry eyes and mouth may
not be entirely the result of dust in the air. His saliva has the
consistency of tile adhesive.
The detachment is so damn secret that no one at the airfield even knows
that they exist. There are a lot of Brits here, and in the desert, Brits
wear shorts, which makes Shaftoe want to punch them in the nose. He controls
the urge. But his obvious hostility towards men in short pants, combined
with the fact that he is demanding to be pointed in the direction of a unit
that is so secret that he cannot specify it by name or even vaguely describe
it, leads to a lot of bafflement, a lot of incredulity, and generally gets
the Anglo American alliance off on the wrong foot.
Sergeant Shaftoe, however, now understands that anything to do with
this detachment is liable to be way off to one side, shrouded in black tarps
and awnings. Like any other military unit, Detachment 2702 is rich in some
supplies and poor in others, but they do appear to control about fifty
percent of last year's total U.S. tarpage production. When Shaftoe mentions
this fact, and goes on about it to his comrades at great length, some of the
men look at him a little funny. It's left to Enoch Root to say, "Between the
giant lizards and the black tarps some people might think you were acting a
little paranoid."
"Let me tell you about paranoid," Shaftoe says, and he does, not
forgetting to mention Lieutenant Ethridge and his wastebaskets. By the time
he's had his say, the whole detachment has assembled on the far side of
those tarps, and everyone is nice and tense except for their newest recruit,
who, as Shaftoe notes approvingly, is beginning to relax. Lying on the bed
of the truck in his wetsuit, he adjusts, rather than bounces, when they go
over bumps.
Even so, he is still stiff enough to simplify the problem of getting
him out of the truck and into their assigned Gooney Bird: a bare knuckled
variant of the DC 3, militarized and (to Shaftoe's skeptical eye) rendered
somewhat less than airworthy by a pair of immense cargo doors gouged into
one side, nearly cutting the airframe in half. This particular Dakota has
been flying around in the fucking desert so long that all the paint's been
sand blasted off its propeller blades, the engine cowling, and the leading
edges of the wings, leaving burnished metal that will make an inviting
silver gleam for any Luftwaffe pilots within three hundred miles. Worse:
diverse antennas sprout from the skin of the fuselage, mostly around the
cockpit. Not just whip antennas but great big damn barbecue grills that make
Shaftoe wish he had a hacksaw. They are eerily like the ones that Shaftoe
humped down the stairway from Station Alpha in Shanghai a memory that has
somehow gotten all mangled together, now, with the other images in his head.
When he tries to recollect it, all he can see is a bloodied Jesus carrying a
high frequency dual band dipole down a stone staircase in Manila, and he
knows that can't be right.
Though they are on the precincts of a busy airfield, Ethridge refuses
to let this operation go forward when there is as much as a single airplane
in the sky. Finally he says, "Okay, NOW!" In the truck, they lift the body
up, just in time to hear Ethridge shout, "No, WAIT!" at which point they put
him down again. Long after it has stopped being grimly amusing, they put a
tarp on Gerald Hott and get him carried on board, and shortly thereafter are
airborne. Detachment 2072 is headed for a rendezvous with Rommel.
Chapter 16 CYCLES
It is early in November of 1942 and a simply unbelievable amount of
shit is going on, all at once, everywhere. Zeus himself would not be able to
sort it all out, not even if he mobilized the caryatids tell them never mind
what we told you, just drop those loads. Temples collapsing everywhere, like
spyglasses, he'd send those caryatids and any naiads and dryads he could
scare up to library school, issue them green visors, dress them in the prim
asexual uniforms of the OPAMS, the Olympian Perspective Archive Management
Service, put them to work filling out three by five cards round the clock.
Get them to use some of that vaunted caryatid steadfastness to tend
Hollerith machines and ETC card readers. Even then, Zeus would probably
still lack a handle on the situation. He'd be so pissed off he would hardly
know which hubristical mortals to fling his thunderbolts at, nor which pinup
girls and buck privates to molest.
Lawrence Pritchard Waterhouse is as Olympian as anyone right now.
Roosevelt and Churchill and the few others on the Ultra Mega list have the
same access, but they have other cares and distractions. They can't wander
around the data flow capital of the planet, snooping over translators'
shoulders and reading the decrypts as they come, chunkity chunkity whirr,
out of the Typex machines. They cannot trace individual threads of the
global narrative at their whim, running from hut to hut patching connections
together, even as the WRENs in Hut 11 string patch cables from one bombe
socket to another, fashioning a web to catch Hitler's messages as they speed
through the ether.
Here are some of the things Waterhouse knows: the Battle of El Alamein
is won, and Montgomery is chasing Rommel westwards across Cyrenaica at what
looks like a breakneck pace, driving him back toward the distant Axis
stronghold of Tunis. But it's not the rout it appears to be. If Monty would
only grasp the significance of the intelligence coming through the Ultra
channel, he would be able to move decisively, to surround and capture large
pockets of Germans and Italians. But he never does, and so Rommel stages an
orderly retreat, preparing to fight another day, and plodding Monty is
roundly cursed in the watch rooms of Bletchley Park for his failure to
exploit their priceless but perishable gems of intelligence.
The largest sealift in history just piled into Northwest Africa. It is
called Operation Torch, and it's going to take Rommel from behind, serving
as anvil to Montgomery's hammer, or, if Monty doesn't pick up the pace a
bit, maybe the other way around. It looks brilliantly organized but it's not
really; this is the first time America has punched across the Atlantic in
any serious way and so a whole grab bag of stuff is included on those ships
including any number of signals intelligence geeks who are storming
theatrically onto the beaches as if they were Marines. Also included in the
landing is the American contingent of Detachment 2702 a hand picked wrecking
crew of combat hardened leathernecks.
Some of these Marines learned what they know on Guadalcanal, a
basically useless island in the Southwest Pacific where the Empire of Nippon
and the United States of America are disputing with rifles each other's
right to build a military airbase. Early returns suggest that the Nipponese
Army, during its extended tour of East Asia, has lost its edge. It would
appear that raping the entire female population of Nanjing, and bayoneting
helpless Filipino villagers, does not translate into actual military
competence. The Nipponese Army is still trying to work out some way to kill,
say, a hundred American Marines without losing, say, five hundred of its own
soldiers.
The Japanese Navy is a different story they know what they are doing.
They have Yamamoto. They have torpedoes that actually explode when they
strike their targets, in stark contrast to the American models which do
nothing but scratch the paint of the Japanese ships and then sink
apologetically. Yamamoto just made another attempt to wipe out the American
fleet off the Santa Cruz Islands, sank Hornet and blew a nice hole in
Enterprise. But he lost a third of his planes. Watching the Japanese rack up
losses, Waterhouse wonders if anyone in Tokyo has bothered to break out the
abacus and run the numbers on this Second World War thing.
The Allies are doing some math of their own, and they are scared
shitless. There are 100 German U boats in the Atlantic now, operating mostly
from Lorient and Bordeaux, and they are slaughtering convoys in the North
Atlantic with such efficiency that it's not even combat, just a Lusitanian
level murder spree. They are on a pace to sink something like a million tons
of shipping this month, which Waterhouse cannot really comprehend. He tries
to think of a ton as being roughly equivalent to a car, and then tries to
imagine America and Canada going out into the middle of the Atlantic and
simply dropping a million cars into the ocean just in November. Sheesh!
The problem is Shark.
The Germans call it Triton. It is a new cypher system, used exclusively
by their Navy. It is an Enigma machine, but not the usual three wheel
Enigma. The Poles learned how to break that old thing a couple of years ago,
and Bletchley Park industrialized the process. But more than a year ago, a
German U boat was beached intact on the south coast of Iceland and gone over
pretty thoroughly by men from Bletchley. They discovered an Enigma box with
niches for four not three wheels.
When the four wheel Enigma had gone into service on February 1st, the
entire Atlantic had gone black. Alan and the others have been going after
the problem very hard ever since. The problem is that they don't know how
the fourth wheel is wired up.
But a few days ago, another U boat was captured, more or less intact,
in the Eastern Mediterranean. Colonel Chattan, who happened to be in the
neighborhood, went there with sickening haste, along with some other
Bletchleyites. They recovered a four wheel Enigma machine, and though this
doesn't break the code, it gives them the data they need to break it.
Hitler must be feeling cocky, anyway, because he's on tour at the
moment, preparatory to a working vacation at his alpine retreat. That didn't
prevent him from taking over what was left of France apparently something
about Operation Torch really got his goat, so he occupied Vichy France in
its entirety, and then dispatched upwards of a hundred thousand fresh
troops, and a correspondingly stupendous amount of supplies, across the
Mediterranean to Tunisia. Waterhouse imagines that you must be able to cross
from Sicily to Tunisia these days simply by hopping from the deck of one
German transport ship to another.
Of course, if that were true, Waterhouse's job would be a lot easier.
The Allies could sink as many of those ships as they wanted to without
raising a single blond Teutonic eyebrow on the information theory front. But
the fact is that the convoys are few and far between. Just exactly how few
and how far between are parameters that go into the equations that he and
Alan Mathison Turing spend all night scribbling on chalkboards.
After a good eight or twelve hours of that, when the sun has finally
come up again, there's nothing like a brisk bicycle ride in the
Buckinghamshire countryside.
***
Spread out before them as they pump over the crest of the rise is a
woods that has turned all of the colors of flame. The hemispherical crowns
of the maples even contribute a realistic billowing effect. Lawrence feels a
funny compulsion to take his hands off the handlebars and clamp them over
his ears. As they coast into the trees, however, the air remains
delightfully cool, the blue sky above unsmudged by pillars of black smoke,
and the calm and quiet of the place could not be more different from what
Lawrence is remembering.
"Talk, talk, talk!" says Alan Turing, imitating the squawk of furious
hens. The strange noise is made stranger by the fact that he is wearing a
gas mask, until he becomes impatient and pulls it up onto his forehead.
"They love to hear themselves talk." He is referring to Winston Churchill
and Franklin Roosevelt. "And they don't mind hearing each other talk up to a
point, at least. But voice is a terribly redundant channel of information,
compared to printed text. If you take text and run it through an Enigma
which is really not all that complicated the familiar patterns in the text,
such as the preponderance of the letter E, become nearly undetectable." Then
he pulls the gas mask back over his face in order to emphasize the following
point: "But you can warp and permute voice in the most fiendish ways
imaginable and it will still be perfectly intelligible to a listener." Alan
then suffers a sneezing fit that threatens to burst the khaki straps around
his head.
"Our ears know how to find the familiar patterns," Lawrence suggests.
He is not wearing a gas mask because (a) there is no Nazi gas attack in
progress, and (b) unlike Alan, he does not suffer from hay fever.
"Excuse me." Alan suddenly brakes and jumps off his bicycle. He lifts
the rear wheel from the pavement, gives it a spin with his free hand, then
reaches down and gives the chain a momentary sideways tug. He is watching
the mechanism intently, interrupted by a few aftersneezes.
The chain of Turing's bicycle has one weak link. The rear wheel has one
bent spoke. When the link and the spoke come into contact with each other,
the chain will part and fall onto the road. This does not happen at every
revolution of the wheel otherwise the bicycle would be completely useless.
It only happens when the chain and the wheel are in a certain position with
respect to each other.
Based upon reasonable assumptions about the velocity that can be
maintained by Dr. Turing, an energetic bicyclist (let us say 25 km/hr) and
the radius of his bicycle's rear wheel (a third of a meter), if the chain's
weak link hit the bent spoke on every revolution, the chain would fall off
every one third of a second.
In fact, the chain doesn't fall off unless the bent spoke and the weak
link happen to coincide. Now, suppose that you describe the position of the
rear wheel by the traditional [theta]. Just for the sake of simplicity, say
that when the wheel starts in the position where the bent spoke is capable
of hitting the weak link (albeit only if the weak link happens to be there
to be hit) then [theta] = 0. If you're using degrees as your unit, then,
during a single revolution of the wheel, [theta] will climb all the way up
to 359 degrees before cycling back around to 0, at which point the bent
spoke will be back in position to knock the chain off And now suppose that
you describe the position of the chain with the variable C, in the following
very simple way: you assign a number to each link on the chain. The weak
link is numbered 0, the next is 1, and so on, up to l – 1 where l is
the total number of links in the chain. And again, for simplicity's sake,
say that when the chain is in the position where its weak link is capable of
being hit by the bent spoke (albeit only if the bent spoke happens to be
there to hit it) then C = 0.
For purposes of figuring out when the chain is going to fall off of Dr.
Turing's bicycle, then, everything we need to know about the bicycle is
contained in the values of [theta] and of C. That pair of numbers defines
the bicycle's state. The bicycle has as many possible states as there can be
different values of ([theta], C) but only one of those states, namely (0,
0), is the one that will cause the chain to fall off onto the road.
Suppose we start off in that state; i.e., with ([theta] = 0, C = 0),
but that the chain has not fallen off because Dr. Turing (knowing full well
his bicycle's state at any given time) has paused in the middle of road
(nearly precipitating a collision with his friend and colleague Lawrence
Pritchard Waterhouse, because his gas mask blocks his peripheral vision).
Dr. Turing has tugged sideways on the chain while moving it forward
slightly, preventing it from being hit by the bent spoke. Now he gets on the
bicycle again and begins to pedal forward. The circumference of his rear
wheel is about two meters, and so when he has moved a distance of two meters
down the road, the wheel has performed a complete revolution and reached the
position [theta] = 0 again that being the position, remember, when its bent
spoke is in position to hit the weak link.
What of the chain? Its position, defined by C, begins at 0 and reaches
1 when its next link moves forward to the fatal position, then 2 and so on.
The chain must move in synch with the teeth on the sprocket at the center of
the rear wheel, and that sprocket has n teeth, and so after a complete
revolution of the rear wheel, when [theta] = 0 again, C = n. After a second
complete revolution of the rear wheel, once again [theta] = 0 but now C =
2n. The next time it's C = 3n and so on. But remember that the chain is not
an infinite linear thing, but a loop having only l positions; at C = l it
loops back around to C = 0 and repeats the cycle. So when calculating the
value of C it is necessary to do modular arithmetic that is, if the chain
has a hundred links (l = 100) and the total number of links that have moved
by is 135, then the value of C is not 135 but 35. Whenever you get a number
greater than or equal to l you just repeatedly subtract l until you get a
number less than 1. This operation is written, by mathematicians, as mod I.
So the successive values of C, each time the rear wheel spins around to
[theta] = 0, are
[C sub i] = n mod l, 2n mod l, 3n mod l,...,in mod l
where i = (1, 2, 3, ... [infinity]) more or less, depending on how
close to infinitely long Turing wants to keep riding his bicycle. After a
while, it seems infinitely long to Waterhouse.
Turing's chain will fall off when his bicycle reaches the state
([theta] = 0, C = 0) and in light of what is written above, this will happen
when (which is just a counter telling how many times the rear wheel has
revolved) reaches some hypothetical value such that in mod l = 0, or, to put
it in plain language, it will happen if there is some multiple of n (such
as, oh, 2n, 3n, 395n or 109,948,368,443n) that just happens to be an exact
multiple of l too. Actually there might be several of these so called common
multiples, but from a practical standpoint the only one that matters is the
first one the least common multiple, or LCM because that's the one that will
be reached first and that will cause the chain to fall off.
If, say, the sprocket has twenty teeth (n 20) and the chain has a
hundred teeth (l 100) then after one turn of the wheel we'll have C 20,
after two turns C = 40, then 60, then 80, then 100. But since we are doing
the arithmetic modulo 100, that value has to be changed to zero. So after
five revolutions of the rear wheel, we have reached the state ([theta] = 0,
C = 0) and Turing's chain falls off. Five revolutions of the rear wheel only
gets him ten meters down the road, and so with these values of l and n the
bicycle is very nearly worthless. Of course, this is only true if Turing is
stupid enough to begin pedaling with his bicycle in the chain falling off
state. If, at the time he begins pedaling, it is in the state ([theta] = 0,
C = 1) instead, then the successive values will be C 21, 41, 61, 81, 1, 21,
. . . and so on forever the chain will never fall off. But this is a
degenerate case, where "degenerate," to a mathematician, means "annoyingly
boring." In theory, as long as Turing put his bicycle into the right state
before parking it outside a building, no one would be able to steal it the
chain would fall off after they had ridden for no more than ten meters.
But if Turing's chain has a hundred and one links (l = 101) then after
five revolutions we have C = 100, and after six we have C = 19, then
C = 39, 59, 79, 99, 18, 38, 58, 78, 98, 17, 37, 57, 77, 97, 16, 36, 56,
76, 96, 15, 35, 55, 75, 95, 14, 34, 54, 74, 94, 13, 33, 53, 73, 93, 12, 32,
52, 72, 92, 11, 31, 51, 71, 91, 10, 30, 50, 70, 90, 9, 29, 49, 69, 89, 8,
28, 48, 68, 88, 7, 27, 47, 67, 87, 6, 26, 46, 66, 86, 5, 25, 45, 65, 85, 4,
24, 44, 64, 84, 3, 23, 43, 63, 83, 2, 22, 42, 62, 82, 1, 21, 41, 61, 81, 0
So not until the 101st revolution of the rear wheel does the bicycle
return to the state ([theta] = 0, C = 0) where the chain falls off. During
these hundred and one revolutions, Turing's bicycle has proceeded for a
distance of a fifth of a kilometer down the road, which is not too bad. So
the bicycle is usable. However, unlike in the degenerate case, it is not
possible for this bicycle to be placed in a state where the chain never
falls off at all. This can be proved by going through the above list of
values of C, and noticing that every possible value of C every single number
from 0 to 100 is on the list. What this means is that no matter what value C
has when Turing begins to pedal, sooner or later it will work its way round
to the fatal C = 0 and the chain will fall off. So Turing can leave his
bicycle anywhere and be confident that, if stolen, it won't go more than a
fifth of a kilometer before the chain falls off.
The difference between the degenerate and nondegenerate cases has to do
with the properties of the numbers involved. The combination of (n = 20, I =
100) has radically different properties from (n = 20, l = 101). The key
difference is that 20 and 101 are "relatively prime" meaning that they have
no factors in common. This means that their least common multiple, their
LCM, is a large number it is, in fact, equal to l x n = 20 x 101 = 2020.
Whereas the LCM of 20 and 100 is only 100. The 101 bicycle has a long period
– it passes through many different states before returning back to the
beginning whereas the l = 100 bicycle has a period of only a few states.
Suppose that Turing's bicycle were a cipher machine that worked by
alphabetic substitution, which is to say that it would replace each of the
26 letters of the alphabet with some other letter. An A in the plaintext
might become a T in the ciphertext, B might become F, C might be come M, and
so on all the way through to Z. In and of itself this would be an absurdly
easy cipher to break kids in treehouses stuff. But suppose that the
substitution scheme changed from one letter to the next. That is, suppose
that after the first letter of the plaintext was enciphered using one
particular substitution alphabet, the second letter of plaintext was
enciphered using a completely different substitution alphabet, and the third
letter a different one yet, and so on. This is called a polyalphabetic
cipher.
Suppose that Turin