The Return of the Black Widowers

  Parris said impatiently, "You are still not serious, gentlemen. None of this is worth anything at all. Please! If you want to keep the money from falling into vile hands, you must do better."

  Avalon, who had had a tight smile on his face for the preceding few minutes, now hunched his magnificent eyebrows down over his eyes and let out a satanic cackle. "But I have it, gentlemen, and I'm delighted to be able to say that Henry, our unexcelled waiter, has overlooked the key clue. No matter, Henry. Even Homer nods."

  "Far less often than I do, Mr. Avalon. What clue did I overlook, sir?"

  "Why, in the preliminary message, there is not only the monogram, as you correctly pointed out, Henry, but also a reference to the fact that m-o-o-d, spelled backward, is d-o-o-m. That statement is rather a non sequitur, and we have a right to wonder why it's brought in at all."

  "Because that's the way Ralph thinks—or thought," said Drake.

  "Undoubtedly, but if you will take the trouble to spell Avalon backward, you have n-o 1-a-v-a. No puns, no rearrangements, just do as Ralph did in the message."

  Parris clenched both hands in excitement. "Now, that's the most interesting thing I've heard yet. But why 'no lava?"

  Avalon said, "A piece of ground over which lava has not flowed is bare."

  Parris considered this and shook his head. "We might just as easily consider that ground over which lava has not flowed is rich in vegetation and is not bare. In that sense, it would be land which lava has flowed that would be bare."

  Avalon said, "Very well, then, we can rearrange the letters slightly and we have o-n 1-a-v-a. By Councilor Parris's argument there would be no vegetation on lava, and that anagram represents bareness."

  "What about the reversed lettering?" said Gonzalo. "Mood to doom and all that." "Well," said Avalon, "we'll have to eliminate that."

  Parris said, "I liked 'no lava,' but it was not convincing. The reason I liked it, though, was that the backward spelling did seem to be a reasonable solution. 'On lava' without the backward spelling has nothing to recommend it."

  There was a moment of silence and Rubin said, "You know, this is getting less funny all the time. Are we going to end up giving the money to the Nazis, even with Henry's help?"

  Gonzalo said, "Well, let's ask him. What are we doing wrong, Henry?"

  Henry said, "I'm not sure, Mr. Gonzalo. It does occur to me, though, that so far we have been punning and anagramming our last names—that is, the potential answers. Ought we to be working the question as well?"

  "I don't see what you mean, Henry," said Avalon.

  "It strikes me, Mr. Avalon, that the phrase 'to the barest' might just possibly be punned into 'to the bearest'; that is, b-e-a-r-e-s-t, the Black Widower most like a bear."

  Trumbull said, "Terrible! It's a terrible pun and it's a terrible suggestion. I don't see how we can get any one of us to be clearly most like a bear anymore than we can get any one of us most bare."

  Gonzalo said, "I don't know, Tom. You've got a terrible temper. You're the most bearish."

  "Not while Manny is alive," said Trumbull hotly.

  "I've never lost my temper in my life, damn it," shouted Rubin, just as hotly.

  "Yes, like now," said Halsted.

  Parris said, "Gentlemen, this is getting us nowhere cither. Unless someone can think of something, we'll have to give up."

  Henry said, "But we now have our solution, to my way of thinking, Mr. Parris. If we take the challenge to be that of finding the Black Widower most like a bear, may I point out that if we change the position of but one letter in r-u-b-i-n, we get b-r-u-i-n, the common name for the bear in the medieval animal epics, and still used today. I believe there is a hockey team known as the 'Bruins.' "

  Parris said energetically, "I'll buy that. It is a clear solution that fits and is unique."

  The Black Widowers broke into applause, and Henry turned pink.

  Rubin said, "Since the money is mine, then, I will set up the trust fund with directions that the earned interest be turned over to Henry as an honorarium for his services to the club."

  There was applause again.

  Henry said, "Gentlemen, please don't. I will be overpaid."

  "Come, come, Henry," said Rubin, "Are you refusing?"

  Henry considered, sighed, and said, "I accept, sir, with thanks."

  Return to Table of Contents

  SIXTY MILLION TRILLION COMBINATIONS

  S

  ince 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 pre-prandial 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."

  709 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 star-tingly 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 you 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 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 1. 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 proo
f 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."