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"Fishing once more? "propelling pencil propelling penciltpropelling pencile 128 .In using this system, absentee voters werepropelling pencil instructed to mark their ballots with number two pencils. The optical scanner rejected ballots which were marked withpropelling pencil instruments other than number two pencil. 2006, Barbara Slater Stern, curriculum and teaching dialogue: V. 8, page 33."Mrs. Landry, I forgot to bring my number two pencils! Does that mean I flunk this test?" Big tears filled Joey's eyes. A standard, ordinary number 2 pencil, one which is not colored or mechan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, propelling pencilhowever, 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, wpropelling pencilere 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 independentpropelling pencilly; 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 propelling pencilwon'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 propelling pencilhave 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 cpropelling pencilall 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."

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