Isaac Asimov - Bouquets of the Black Widowers

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BOUQUETS OF THE BLACK WIDOWERS

ISAAC ASIMOV, 1984

Sixty Million Trillion Combinations

SINCE IT WAS Thomas Trumbull who was going to act as host for the Black Widowers that month, he

did not, as was his wont, arrive at the last minute, gasping for his preprandial drink.

There he was, having arrived in early dignity, conferring with Henry, that peerless waiter, on the details of

the menu for the evening, and greeting each of the others as he arrived.

Mario Gonzalo, who arrived last, took off his light overcoat with care, shook it gently, as though to

remove the dust of the taxicab, and hung it up in the cloakroom. He came back, rubbing his hands, and

said, "There's an autumn chill in the air. I think summer's over."

"Good riddance," called out Emmanuel Rubin, from where he stood conversing with Geoffrey Avalon

and James Drake.

"I'm not complaining," called back Gonzalo. Then, to Trumbull, "Hasn't your guest arrived yet?"

Trumbull said distinctly, as though tired of explaining, "I have not brought a guest."

"Oh?" said Gonzalo, blankly. There was nothing absolutely irregular about that. The rules of the Black

Widowers did not require a guest, although not to have one was most unusual. "Well, I guess that's all

right."

"It's more than all right," said Geoffrey Avalon, who had just drifted in their direction, gazing down from

his straight - backed height of seventy - four inches. His thick graying eyebrows hunched over his eyes

and he said, "At least that guarantees us one meeting in which we can talk aimlessly and relax."

Gonzalo said, "I don't know about that. I'm used to the problems that come up. I don't think any of us

will feel comfortable without one. Besides, what about Henry?"

He looked at Henry as he spoke and Henry allowed a discreet smile to cross his unlined, sixtyish face.

"Please don't be concerned, Mr. Gonzalo. It will be my pleasure to serve the meal and attend the

conversation even if there is nothing of moment to puzzle us."

"Well," said Trumbull, scowling, his crisply waved hair startlingly white over his tanned face, "you won't

have that pleasure, Henry. I'm the one with the problem and I hope someone can solve it: you at least,

Henry."

Avalon's lips tightened, "Now by Beelzebub's brazen bottom, Tom, you might have given us one old -

fashioned - "

Trumbull shrugged and turned away, and Roger Halsted said to Avalon in his soft voice, "What's that

Beelzebub bit? Where'd you pick that up?"

Avalon looked pleased. "Oh, well, Manny is writing some sort of adventure yarn set in Elizabeth's

England - Elizabeth I of course - and it seems - "

Rubin, having heard the magic sound of his name, approached and said, "It's a sea story."

Halsted said, "Are you tired of mysteries?"

"It's a mystery also," said Rubin, his eyes flashing behind the thick lenses of his glasses. "What makes you

think you can't have a mystery angle to any kind of story?"

"In any case," said Avalon, "Manny has one character forever swearing alliteratively and never the same

twice and he needs a few more resounding oaths. Beelzebub's brazen bottom is good, I think."

"Or Mammon's munificent mammaries," said Halsted.

Trumbull said, violently, "There you are! If I don't come up with some problem that will occupy us in

worthwhile fashion and engage our Henry's superlative mind, the whole evening would degenerate into

stupid triplets - by Tutankhamen's tin trumpet."

"It gets you after a while," grinned Rubin, unabashed.

"Well, get off it," said Trumbull. "Is dinner ready, Henry?"

"Yes it is, Mr. Trumbull."

"All right, then. If you idiots keep this alliteration up for more than two minutes, I'm walking out, host or

no host."

The table seemed empty with only six about it, and conversation seemed a bit subdued with no guest to

sparkle before.

Gonzalo, who sat next to Trumbull, said, "I ought to draw a cartoon of you for our collection since you're

your own guest, so to speak." He looked up complacently at the long list of guest - caricatures that lined

the wall in rank and file. "We're going to run out of space in a couple of years."

"Then don't bother with me," said Trumbull, sourly, "and we can always make space by burning those

foolish scrawls."

"Scrawls!" Gonzalo seemed to debate within himself briefly concerning the possibility of taking offense.

Then he compromised by saying, "You seem to be in a foul mood, Tom."

"I seem so because I am. I'm in the situation of the Chaldean wise men facing Nebuchadnezzar."

Avalon leaned over from across the table. "Are you talking about the Book of Daniel, Tom?"

"That's where it is, isn't it?"

Gonzalo said, "Pardon me, but I didn't have my Bible lesson yesterday. What are these wise men?"

"Tell him, Jeff," said Trumbull. "Pontificating is your job."

Avalon said, "It's not pontificating to tell a simple tale. If you would rather - "

Gonzalo said, "I'd rather you did, Jeff. You do it much more authoritatively."

"Well," said Avalon, "it's Rubin, not I, who was once a boy preacher, but I'll do my poor best. The

second chapter of the Book of Daniel tells that Nebuchadnezzar was once troubled by a bad dream and

he sent for his Chaldean wise men for an interpretation. The wise men offered to do so at once as soon

as they heard the dream but Nebuchadnezzar couldn't remember the dream, only that he had been

disturbed by it. He reasoned, however, that if wise men could interpret a dream, they could work out the

dream, too, so he ordered them to tell him both the dream and the interpretation. When they couldn't do

this, he very reasonably - by the standards of Oriental potentates - ordered them all killed. Fortunately

for them Daniel, a captive Jew in Babylon, could do the job."

Gonzalo said, "And that's your situation, too, Tom?"

"In a way. I have a problem that involves a cryptogram - but I don't have the cryptogram. I have to work

out the cryptogram."

"Or you'll be killed?" asked Rubin.

"No. If I fail, I won't be killed, but it won't do me any good, either."

Gonzalo said, "No wonder you didn't feel it necessary to bring a guest. Tell us all about it."

"Before the brandy?" said Avalon, scandalized.

"Tom's host," said Gonzalo, defensively. "If he wants to tell us now - "

"I don't," said Trumbull. "We'll wait for the brandy as we always do, and I'll be my own griller, if you

don't mind."

When Henry was pouring the brandy, Trumbull rang his spoon against his water glass and said,

"Gentlemen, I will dispense with the opening question by admitting openly that I cannot justify my

existence. Without pretending to go on by question - and - answer, I will simply state the problem. You

are free to ask questions, but for God's sake, don't get me off on any wild - goose chases. This is

serious."

Avalon said, "Go ahead, Tom. We will do our best to listen."

Trumbull said, with a certain weariness, "It involves a fellow named Pochik. I've got to tell you a little

about him in order to let you understand the problem but, as is usual in these cases, I hope you don't

mind if I tell you nothing that isn't relevant.

"In the first place he's from Eastern Europe, from someplace in Slovenia, I think, and he came here at

about fourteen. He taught himself English, went to night school and to University Extension, working

every step of the way. He worked as a waiter for ten years, while he was taking his various courses, and

you know what that means. Sorry, Henry."

Henry said, tranquilly, "It is not necessarily a pleasant occupation. Not everyone waits on the Black

Widowers, Mr. Trumbull."

"Thank you, Henry. That's very diplomatic of you. However, he wouldn't have made it, if it weren't plain

from the start that he was a mathematical wizard. He was the kind of young man that no mathematics

professor in his right mind wouldn't have moved heaven and earth to keep in school. He was their claim

to a mark in the history books - that they had taught Pochik. Do you understand?"

Avalon said, "We understand, Tom."

Trumbull said, "At least, that's what they tell me. He's working for the government now, which is where I

come in. They tell me he's something else. They tell me he's in a class by himself. They tell me he can do

things no one else can. They tell me they've got to have him. I don't even know what he's working on, but

they've got to have him."

Rubin said, "Well, they've got him, haven't they? He hasn't been kidnapped and hijacked back across the

Iron Curtain, has he?"

"No, no," said Trumbull, "nothing like that. It's a lot more irritating. Look, apparently a great

mathematician can be an idiot in every other respect."

"Literally an idiot?" asked Avalon. "Usually idiots savants have remarkable memories and can play

remarkable tricks in computation, but that is far from being any kind of mathematician, let alone a great

one."

"No, nothing like that, either." Trumbull was perspiring and paused to mop at his forehead. "I mean he's

childish. He's not really learned in anything but mathematics and that's all right. Mathematics is what we

want out of him. The trouble is that he feels backward; he feels stupid. Damn it, he feels inferior, and

when he feels too inferior, he stops working and hides in his room."

Gonzalo said, "So what's the problem? Everyone just has to keep telling him how great he is all the time."

"He's dealing with other mathematicians and they're almost as crazy as he is. One of them, Sandino, hates

being second best and every once in a while he gets Pochik into a screaming fit. He's got a sense of

humor, this Sandino, and he likes to call out to Pochik, 'Hey, waiter, bring the check.' Pochik can't ever

learn to take it."

Drake said, "Read this Sandino the riot act. Tell him you'll dismember him if he tries anything like that

again."

"They did," said Trumbull, "or at least as far as they quite dared to. They don't want to lose Sandino

either. In any case, the horseplay stopped but something much worse happened. You see there's

something called, if I've got it right, 'Goldbach's conjecture'."

Roger Halsted galvanized into a position of sharp interest at once. "Sure," he said. "Very famous."

"You know about it?" said Trumbull.

Halsted stiffened. "I may just teach algebra to junior high school students, but yes, I know about

Goldbach's conjecture. Teaching a junior high school student doesn't make me a junior - "

"All right. I apologize. It was stupid of me," said Trumbull. "And since you're a mathematician, you can be

temperamental too. Anyway, can you explain Goldbach's conjecture? - Because I'm not sure I can."

"Actually," said Halsted, "it's very simple. Back in 1742, I think, a Russian mathematician, Christian

Goldbach, stated that he believed every even number greater than 2 could be written as the sum of two

primes, where a prime is any number that can't be divided evenly by any other number but itself and i.

For instance, 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 10 = 3 + 7; 12 = 5 + 7; and so on, as far as you want to

go."

Gonzalo said, "So what's the big deal?"

"Goldbach wasn't able to prove it. And in the two hundred and something years since his time, neither

has anyone else. The greatest mathematicians haven't been able to show that it's true."

Gonzalo said, "So?"

Halsted said patiently, "Every even number that has ever been checked always works out to be the sum

of two primes. They've gone awfully high and mathematicians are convinced the conjecture is true - but

no one can prove it."

Gonzalo said, "If they can't find any exceptions, doesn't that prove it?"

"No, because there are always numbers higher than the highest we've checked, and besides we don't

know all the prime numbers and can't, and the higher we go, then the harder it is to tell whether a

particular number is prime or not. What is needed is a general proof that tells us we don't have to look

for exceptions because there just aren't any. It bothers mathematicians that a problem can be stated so

simply and seems to work out, too, and yet that it can't be proved."

Trumbull had been nodding his head. "All right, Roger, all right. We get it. But tell me, does it matter?

Does it really matter to anyone who isn't a mathematician whether Goldbach's conjecture is true or not;

whether there are any exceptions or not?"

"No," said Halsted. "Not to anyone who isn't a mathematician; but to anyone who is and who manages

either to prove or disprove Goldbach's conjecture, there is an immediate and permanent niche in the

mathematical hall of fame."

Trumbull shrugged. "There you are. What Pochik's really doing is of great importance. I'm not sure

whether it's for the Department of Defense, the Department of Energy, NASA, or what, but it's vital.

What he's interested in, however, is Goldbach's conjecture, and for that he's been using a computer."

"To try higher numbers?" asked Gonzalo.

Halsted said promptly, "No, that would do no good. These days, though, you can use computers on

some pretty recalcitrant problems. It doesn't yield an elegant solution, but it is a solution. If you can

reduce a problem to a finite number of possible situations - say, a million - you can program a computer

to try every one of them. If every one of them checks out as it's supposed to, then you have your proof.

They recently solved the four - color mapping problem that way; a problem as well known and as

recalcitrant as Goldbach's conjecture."

"Good," said Trumbull, "then that's what Pochik's been doing. Apparently, he had worked out the

solution to a particular lemma. Now what's a lemma?"

Halsted said, "It's a partway solution. If you're climbing a mountain peak and you set up stations at

various levels, the lemmas are analogous to those stations and the solution to the mountain peak."

"If he solves the lemma, will he solve the conjecture?"

"Not necessarily," said Halsted, "any more than you'll climb the mountain if you reach a particular station

on the slopes. But if you don't solve the lemma, you're not likely to solve the problem, at least not from

that direction."

"All right, then," said Trumbull, sitting back. "Well, Sandino came up with the lemma first and sent it in for

publication."

Drake was bent over the table, listening closely. He said, "Tough luck for Pochik."

Trumbull said, "Except that Pochik says it wasn't luck. He claims Sandino doesn't have the brains for it

and couldn't have taken the steps he did independently; that it is asking too much of coincidence."

Drake said, "That's a serious charge. Has Pochik got any evidence?"

"No, of course not. The only way that Sandino could have stolen it from Pochik would have been to tap

the computer for Pochik's data and Pochik himself says Sandino couldn't have done that."

"Why not?" said Avalon.

"Because," said Trumbull, "Pochik used a code word. The code word has to be used to alert the

computer to a particular person's questioning. Without that code word, everything that went in with the

code word is safely locked away."

Avalon said, "It could be that Sandino learned the code word."

"Pochik says that is impossible," said Trumbull. "He was afraid of theft, particularly with respect to

Sandino, and he never wrote down the code word, never used it except when he was alone in the room.

What's more, he used one that was fourteen letters long, he says. Millions of trillions of possibilities, he

says. No one could have guessed it, he says."

Rubin said, "What does Sandino say?"

"He says he worked it out himself. He rejects the claim of theft as the ravings of a madman. Frankly, one

could argue that he's right."

Drake said, "Well, let's consider. Sandino is a good mathematician and he's innocent till proven guilty.

Pochik has nothing to support his claim and Pochik actually denies that Sandino could possibly have

gotten the code word, which is the only way the theft could possibly have taken place. I think Pochik has

to be wrong and Sandino right."

Trumbull said, "I said one could argue that Sandino's right, but the point is that Pochik won't work. He's

sulking in his room and reading poetry and he says he will never work again. He says Sandino has

robbed him of his immortality and life means nothing to him without it."

Gonzalo said, "If you need this guy so badly can you talk Sandino into letting him have his lemma?"

"Sandino won't make the sacrifice and we can't make him unless we have reason to think that fraud was

involved. If we get any evidence to that effect we can lean on him hard enough to squash him flat. - But

now listen, I think it's possible Sandino did steal it."

Avalon said, "How?"

"By getting the code word. If I knew what the code word was, I'm sure I could figure out a logical way in

which Sandino could have found it out or guessed it. Pochik, however simply won't let me have the code

word. He shrieked at me when I asked. I explained why, but he said it was impossible. He said Sandino

did it some other way - but there is no other way."

Avalon said, "Pochik wants an interpretation but he won't tell you the dream, and you have to figure out

the dream first and then get the interpretation."

"Exactly! Like the Chaldean wise men."

"What are you going to do?"

"I'm going to try to do what Sandino must have done. I'm going to try to figure out what the fourteen -

letter code word was and present it to Pochik. If I'm right, then it will be clear that what I could do,

Sandino could do, and that the lemma was very likely stolen."

There was a silence around the table and then Gonzalo said, "Do you think you can do it, Tom?"

"I don't think so. That's why I've brought the problem here. I want us all to try. I told Pochik I would call

him before 10:30 P.M. tonight" - Trumbull looked at his watch - "with the code word just to show him it

could be broken. I presume he's waiting at the phone."

Avalon said, "And if we don't get it?"

"Then we have no reasonable way of supposing the lemma was stolen and no really ethical way of trying

to force it away from Sandino. But at least we'll be no worse off."

Avalon said, "Then you go first. You've clearly been thinking about it longer than we have, and it's your

line of work."

Trumbull cleared his throat. "All right. My reasoning is that if Pochik doesn't write the thing down, then

he's got to remember it. There are some people with trick memories and such a talent is fairly common

among mathematicians. However, even great mathematicians don't always have the ability to remember

long strings of disjointed symbols and, upon questioning of his coworkers, it would seem quite certain

that Pochik's memory is an ordinary one. He can't rely on being able to remember the code unless it's

easy to remember.

"That would limit it to some common phrase or some regular progression that you couldn't possibly

forget. Suppose it were ALBERT EINSTEIN, for instance. That's fourteen letters and there would be no

fear of forgetting it. Or SIR ISAAC NEWTON, or ABCDEFGHIJKLMN, or, for that matter,

NMLKJIHGFEDCBA. If Pochik tried something like this, it could be that Sandino tried various obvious

combinations and one of them worked."

Drake said, "If that's true, then we haven't a prayer of solving the problem. Sandino might have tried any

number of different possibilities over a period of months. One of them finally worked. If he got it by hit -

and - miss over a long time, we have no chance in getting the right one in an hour and a half, without even

trying any of them on the computer."

"There's that, of course," said Trumbull, "and it may well be that Sandino had been working on the

problem for months. Sandino pulled the waiter routine on Pochik last June, and Pochik, out of his mind,

screamed at him that he would show him when his proof was ready. Sandino may have put this together

with Pochik's frequent use of the computer and gotten to work. He may have had months, at that."

"Did Pochik say something on that occasion that gave the code word away?" asked Avalon.

"Pochik swears all he said was "I'll show you when the proof is ready,' but who knows? Would Pochik

remember his own exact words when he was beside himself?"

Halsted said, "I'm surprised that Pochik didn't try to beat up this Sandino."

Trumbull said, "You wouldn't be surprised if you knew them. Sandino is built like a football player and

Pochik weighs 110 pounds with his clothes on."

Gonzalo said, suddenly, "What's this guy's first name?"

Trumbull said, "Vladimir."

Gonzalo paused a while, with all eyes upon him, and then he said, "I knew it. VLADIMIR POCHIK has

fourteen letters. He used his own name."

Rubin said, "Ridiculous. It would be the first combination anyone would try."

"Sure, the purloined letter bit. It would be so obvious that no one would think to use it. Ask him."

Trumbull shook his head. "No. I can't believe he'd use that."

Rubin said, thoughtfully, "Did you say he was sitting in his room reading poetry?"

"Yes."

"Is that a passion of his? Poetry? I thought you said that outside mathematics he was not particularly

educated."

Trumbull said, sarcastically, "You don't have to be a Ph.D. to read poetry."

Avalon said, mournfully, "You would have to be an idiot to read modern poetry."

"That's a point," said Rubin. "Does Pochik read contemporary poetry?"

Trumbull said, "It never occurred to me to ask. When I visited him, he was reading from a book of

Wordsworth's poetry, but that's all I can say."

"That's enough," said Rubin. "If he likes Wordsworth then he doesn't like contemporary poetry. No one

can read that fuddy - duddy for fun and like the stuff they turn out these days."

"So? What difference does it make?" asked Trumbull.

"The older poetry with its rhyme and rhythm is easy to remember and it could make for code words. The

code word could be a fourteen - letter passage from one of Wordsworth's poems, possibly a common

one: LONELY AS A CLOUD has fourteen letters. Or any fourteen - letter combinations from such lines

as "The child is father of the man' or 'trailing clouds of glory' or 'Milton! thou shouldst be living at this

hour.' Or maybe from some other poet of the type."

Avalon said, "Even if we restrict ourselves to passages from the classic and romantic poets, that's a huge

field to guess from."

Drake said, "I repeat. It's an impossible task. We don't have the time to try them all. And we can't tell

one from another without trying."

Halsted said, "It's even more impossible than you think, Jim. I don't think the code word was in English

words."

Trumbull said, frowning, "You mean he used his native language?"

"No, I mean he used a random collection of letters. You say that Pochik said the code word was

unbreakable because there were millions of trillions of possibilities in a fourteen - letter combination.

Well, suppose that the first letter could be any of the twenty - six, and the second letter could be any of

the twenty - six, and the third letter, and so on. In that case the total number of combinations would be

26 X 26 X 26, and so on. You would have to get the product of fourteen 26's multiplied together and the

result would be" - he took out his pocket calculator and manipulated it for a while - "about 64 million

trillion different possibilities.

"Now, if you used an English phrase or a phrase in any reasonable European language, most of the letter

combinations simply don't occur. You're not going to have an HGF or a QXZ or an LLLLC. If we

include only possible letter combinations in words then we might have trillions of possibilities, probably

less, but certainly not millions of trillions. Pochik, being a mathematician, wouldn't say millions of trillions

unless he meant exactly that, so I expect the code word is a random set of letters."

Trumbull said, "He doesn't have the kind of memory - "

Halsted said, "Even a normal memory will handle fourteen random letters if you stick to it long enough."

Gonzalo said, "Wait awhile. If there are only so many combinations, you could use a computer. The

computer could try every possible combination and stop at the one that unlocks it."

Halsted said, "You don't realize how big a number like 64 million trillion really is, Mario. Suppose you

arranged to have the computer test a billion different combinations every second. It would take two

thousand solid years of work, day and night, to test all the possible combinations."

Gonzalo said, "But you wouldn't have to test them all. The right one might come up in the first two hours.

Maybe the code was AAAAAAAAAAAAAA and it happened to be the first one the computer tried."

"Very unlikely," said Halsted. "He wouldn't use a solid - A code anymore than he would use his own

name. Besides Sandino is enough of a mathematician not to start a computer attempt he would know

could take a hundred lifetimes."

Rubin said, thoughtfully, "If he did use a random code, I bet it wasn't truly random."

Avalon said, "How do you mean, Manny?"

"I mean if he doesn't have a superlative memory and he didn't write it down, how could he go over and

over it in his mind in order to memorize it? Just repeat fourteen random letters to yourself and see if you

can be confident of repeating them again in the exact order immediately afterward. And even if he had

worked out a random collection of letters and managed to memorize it, it's clear he had very little self -

confidence in anything except mathematical reasoning. Could he face the possibility of not being able to

retrieve his own information because he had forgotten the code?"

"He could start all over," said Trumbull.

"With a new random code? And forget that, too?" said Rubin. "No. Even if the code word seems

random, I'll bet Pochik has some foolproof way of remembering it, and if we can figure out the foolproof

way, we'd have the answer. In fact, if Pochik would give us the code word, we'd see how he memorized

it and then see how Sandino broke the code."

Trumbull said, "And if Nebuchadnezzar would only have remembered the dream, the wise men could

have interpreted it. Pochik won't give us the code word, and if we work it with hindsight, we'll never be

sufficiently sure Sandino cracked it without hindsight. - All right, we'll have to give it up."

"It may not be necessary to give it up," said Henry, suddenly. "I think - "

All turned to Henry, expectantly. "Yes, Henry," said Avalon. "I have a wild guess. It may be all wrong.

Perhaps it might be possible to call up Mr. Pochik, Mr. Trumbull, and ask him if the code word is

WEALTMDITEBIAT," said Henry.

Trumbull said, "What?"

Halsted said, his eyebrows high, "That's some wild guess, all right. Why that?"

Gonzalo said, "It makes no sense."

No one could recall ever having seen Henry blush, but he was distinctly red now. He said, "If I may be

excused. I don't wish to explain my reasoning until the combination is tried. If I am wrong, I would

appear too foolish. And, on second thought, I don't urge it be tried."

Trumbull said, "No, we have nothing to lose. Could you write down that letter combination, Henry?"

"I have already done so, sir."

Trumbull looked at it, walked over to the phone in the corner of the room, and dialled. He waited for

four rings, which could be clearly heard in the breath - holding silence of the room. There was then a

click, and a sharp, high - pitched "Hello?"

Trumbull said, "Dr. Pochik? Listen. I'm going to read some letters to you - No, Dr. Pochik, I'm not

saying I've worked out the code. This is an exper - It's an experiment sir. We may be wrong - No, I

can't say how - Listen, W, E, A, L - Oh, good God." He placed his hand over the mouthpiece. "The man

is having a fit."

"Because it's right or because it's wrong?" asked Rubin.

"I don't know." Trumbull put the phone back to his ear. "Dr. Pochik, are you there? - Dr. Pochik? - The

rest is" - he consulted the paper - "T, M, D, I, T, E, B, I, A, T." He listened. "Yes, sir, I think Sandino

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BOUQUETSOFTHEBLACKWIDOWERSISAACASIMOV,1984SixtyMillionTrillionCombinationsSINCEITWASThomasTrumbullwhowasgoingtoactashostfortheBlackWidowersthatmonth,hedidnot,aswashiswont,arriveatthelastminute,gaspingforhispreprandialdrink.Therehewas,havingarrivedinearlydignity,conferringwithHenry,thatpeerlesswaiter...

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Isaac Asimov - Bouquets of the Black Widowers

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