Mario Gonzalo, the artist member of the Black Widowers, seemed oddly disheveled as he said vehemently, "I cannot teach what I do because I don't know what I do, but that doesn't mean I can't do it."

And Emmanuel Rubin, his straggly beard seeming to shoot sparks out of each gray bristle, bent his eyes, magnified through the thick lenses before them, at Gonzalo and said, "If you don't know what you're doing, you're a paint dauber and not an artist."

"You're a madman, Manny. If knowing were everything, Michelangelo could teach you to be Michelangelo, but the fact is that Michelangelo couldn't teach anyone to be Michelangelo. For that matter, no one could have taught Michelangelo to be Michelangelo either. He was born Michelangelo."

"You miss the entire point. Teaching doesn't necessarily imply the making of an equal. Michelangelo could give the kind of instruction from which others might profit. If he couldn't create equals, he could create somewhat less miserable marble tappers. You bet he knew what he was doing even if he could only pound a limited amount into the heads of mere mortals."

"Ah," said Gonzalo gleefully, "Mere mortals! And what made them mere mortals? The lack of genius. And what were the components of that genius? Could Michelangelo himself know?"

Thomas Trumbull, staring over his scotch and soda, and apparently irritated at being excluded from a conversation that the loud voices of Gonzalo and Rubin had made into a private dialog, scowled and said, "Since Michelangelo is dead and can't be consulted on the subject, why don't we drop this foolish argument?"

"No," said Gonzalo passionately, "I appeal from the sublime to the ridiculous and ask Manny. You're a writer, Manny - after a fashion. Can you teach what you do?"

"I not only can," said Rubin, "I have. I've written articles for The Writer and I've lectured at writers' conferences."

"And you've told them about query letters, and the necessity of rewriting, I suppose. Do you tell them how you know where you start your story, just which incident you put after which, how you break up your dialog, how you make the denouement inevitable without giving it away?"

"I could do that."

"Then do it right now. Explain it to me!"

Roger Halsted, flushing to the roots of his receding line of hair, said, in his soft voice, "Don't do it, Manny. We'll be sitting here all night and none of us is interested. Not even Mario."

"I won't - but I can."

"You can't," said Gonzalo, "because you can't describe the intuition involved. Enough intuition is talent, and a hell of a lot of it is genius, and intuition can't be taught."

Geoffrey Avalon, standing tall, said in his solemn baritone, "You stand with the Greeks, Mario. They were quite certain that any outstanding ability was the result of divine inspiration, the working of a god who possessed the person. The word 'enthusiasm,' expressing this process, means 'the god within' in Greek. Naturally, one can't explain the workings of a god to a mere mortal, and that, I take it, is your position, Mario."

"Bull!" said Rubin. "Bull to you, Geoff, and to Mario and to the Greeks. There is nothing mysterious about intuition at all."

"If you can understand it," said Mario, "explain it."

"I will," said Rubin. "All a man knows is what he observes and learns. There is nothing innate except a few biological instincts - certainly nothing cultural. It may be that with experience - with experience, damn you, Mario - a person learns to interpret, very rapidly, what he observes, or to draw inferences, or to do something based on deduction or induction from those observations and his past experience. He does it so rapidly that he generally doesn't bother to isolate the step in the procedure or even to be aware they exist, so he calls it intuition. Yes, Henry?"

Henry, the perennial waiter at the Black Widowers' monthly dinner meetings, his bland and uncreased sixtyish face displaying no emotion, said, gently, "Dinner is served, Mr. Rubin. It you will sit down, I am sure the rest will follow."

Rubin said, "I suppose I am the natural leader."

"No," said James Drake, stubbing out his cigarette, "as host today, I'm the leader. However, the rest of us are naturally afraid you'll eat everything in sight if we don't all sit down to protect our rights."

"That depends," said Rubin, "on what we are having today. Henry?"

Henry said, "The chef is in an Old English mood today and it will be rib roast and Yorkshire pudding, preceded by a seafood quiche."

"That's not Old English," said Rubin.

"The chef is rarely entirely consistent," said Henry, "and I'm afraid his judgment of what will constitute a success for dinner is largely intuitive."

"And largely right, too," said Gonzalo, approving. "Whatever you say intuition is, Manny, some people have more of it than others, and why is that?"

"Some people have more talent than . . ."

"Aha!" said Gonzalo.

Rubin looked haughty and said with stiff politeness, "If I am to be allowed to finish my sentence, I will go on to explain that talent is the capacity for such fast thought, plus, perhaps, muscular deftness, and undoubtedly depends on the physiology of the brain and on nothing more mysterious than that."

"That's mysterious enough," said Drake.

"Mysterious now, but not necessarily forever," said Rubin, "and when we learn enough about the brain, talent and genius will be as nonmysterious as eye color."

"That's just your intuitive guess," shot back Gonzalo.

Rubin's reply was lost in quiche, and the dinner conversation grew more general.

Through all the argument, the guest of the evening had maintained a steady and clearly amused silence. Quietly, he listened and as quietly, he sipped at his martini.

His name was Simon Alexander. His black hair and black mustache, each thick and luxuriant enough to give him a Satanic appearance, or, failing that, a Levantine one, were his most prominent features. The small and persistent smile on his face seemed to accentuate the Satanism.

When the coffee was served, however, and Drake tapped his spoon against the water glass, Alexander, as though in anticipation, grew serious.

Drake said, "Gentlemen, it is time to grill our esteemed guest, and Manny, since you've been clattering away more insupportably even than usual, suppose you supervise the grilling."

Rubin said, "I'm sorry you find mental stimulation insupportable, Jim, but I'm not surprised." He took a quick sip at his coffee, signaled Henry for a bit of a freshener, then said,

"Well, Mr. Alexander, or, if you prefer, Simon, how do you justify your existence?"

Alexander's smile returned. "By seeing to it that the American people pay their legal taxes in full and on time."

There was a stir about the table, and even Henry was betrayed into pausing in the precise performance of his duties long enough to cast a penetrating glance at the guest.

Trumbull said, with a distinct suggestion of outrage, "Are you in the employ of the IRS?"

"I am," said Alexander. "I'm in the Division of Fraud."

"Good God," said Trumbull, "and you offer that as justification for your existence? Horse - whipping with barbed wire is what it justifies." He cast a lowering glance at Drake.

Drake said, "Give him a chance, Tom. It takes all kinds to make a world and, aside from his profession, Simon is one of Nature's noblemen."

Alexander waved his hand. "It's all right, Jim. Tax collectors have always been the favorite villains of humanity from the moment they first appeared on scene in ancient Sumeria five thousand years ago and invented writing in order to keep score. Besides, I think Mr. Trumbull was merely expressing himself colorfully and didn't really mean it."

"The hell I didn't," muttered Trumbull.

Rubin, who had held his peace in a markedly aggrieved manner, now raised his voice, "Since I'm grilling today, may I continue? Do you mind keeping quiet, Tom?"

Trumbull said, "Circumstances moved me."

Rubin waited for silence and said, "Mr. Alexander - I withdraw the Simon, since through the common law of humanity, you can have no friends here, or possibly anywhere - how can your role in tax collection be taken as justifying your existence?"

Alexander said, "I think it is not difficult to see that the IRS represents the single essential arm of the government. Presidents can die and be replaced at once, with but a slight and perfunctory hiccup of emotion. Congress can bumble, the Supreme Court can drag, and we can lose ground diplomatically, economically even militarily, yet perhaps make it all up afterward. Natural disasters are local, temporary, and pass by.

"However, let the tax structure of the nation falter and the government can no longer function. That would mean a spreading paralysis far wider and longer and deeper than anything that can possibly occur short of a thermonuclear war."

Rubin said, "But the tax structure is not likely to falter, is it?"

"Not in the sense that the physical machinery is likely to break apart or that the computers will stop working. No, the weak link is the taxpayer himself. The American budget now approaches half a trillion dollars annually, and the largest part of this is collected out of the unwilling wallets of Americans everywhere."

"Sorry, Manny," said Trumbull angrily, "but I've got to interrupt. What the hell has 'unwilling' got to do with it? You enforce your own interpretation of the rules, act as prosecuting attorney and judge, hound us relentlessly, treat us as guilty till we prove ourselves innocent, and are perfectly ready to jail us if you can. What do you care if we're unwilling?"

"In the first place," said Alexander, "our judgments can be appealed to the courts. We are not the last word. Second, it would be much more harmful if we were not relentless. Despite everything we do, we cannot audit everyone, we cannot check into everything. If we tried, the cost would far outweigh what additional money we could collect. No, we are forced to depend on the average American filling out a reasonably honest return and we can count on this only as long as that average American is convinced of the essential honesty of the system. Within the bounds of the law - and the law is not completely equitable, but that's not our fault - we must show neither favor nor mercy or the structure will break down.

"Thus, although Al Capone could commit theft on a grand scale and could murder with impunity, he could be grabbed on income - tax evasion. There is nothing ironical in this. Income - tax evasion is the greatest of these. Similarly, nothing that Nixon and Agnew did prior to their forced resignations was as mischievous as their tampering with the IRS and their making out of dubious income - tax returns. That they were willing to shake the faith of the American people in the honesty of the tax structure was of all their misdeeds the most unforgivable."

Rubin said, "You're serious about this now? You're not pulling our legs?"

"Dead serious."

"Good God, Jim," Rubin said, "we ought to ask for your resignation. You've brought in a guy who's going to make it difficult for me to indulge in a little bit of honest expense - padding next time around."

Avalon cleared his throat. "I don't consciously pad, but I must admit that the IRS and I might not agree on just what constitutes a deductible expense in the first place."

"Then you deduct it till we tell you otherwise," said Alexander, agreeably. "That's the tax man's version of keeping you innocent till proven guilty - but none of this is what I came here prepared to talk about."

"Oh," said Rubin, "what are you prepared for?"

"Jim told me," said Alexander, "that the Black Widowers like to hear some tale that involves a bit of a puzzle, and I happen to have one."

"Jim was wrong to tell you that," said Avalon, austerely. "We meet for the purpose of participating in stimulating conversation, and a puzzle is not necessary; however -"

Alexander smiled. "In that connection, I was amused by the preprandial quarrel over the nature of intuition, since it is with a matter of intuition that my story is concerned."

"Telepathy!" said Gonzalo at once.

"No, I think not," said Alexander. 'The whole conversation illustrated Mr. Rubin's thesis, actually. I agree with him that intuition is undetected observation and deduction, and I would like to point out that what is often considered telepathy is the same. Thus, when Jim introduced me, he said - and I think these are his exact words  -  'This is Simon Alexander, an investigator of sorts and a very good one. He can actually sense criminality by some kind of inner magic, I think.' Isn't that what you said, Jim?"

"I think so," said Drake.

"I notice you said, 'an investigator of sorts,'" growled Trumbull. "You didn't say he was from the IRS."

Alexander said, "I am trying to make the point that the introduction took place when the drinks were being passed around - as the brandy is now, I see. Henry, do we have any curacao?"

"I believe so, sir."

"I'll have some then. With everyone concentrating on alcohol, I don't think anyone heard the introduction. Does anyone recall having heard it?"

There were no bites on that one, and Alexander smoothed his mustache with one forefinger and accepted the small glass with its orange - colored content from Henry.

"But it Rubin and Gonzalo did not consciously hear it," he said, "they nevertheless heard it, I'm convinced, and that bit about sensing criminality by some kind of inner magic sparked the argument on intuition. Of course, I don't use some kind of inner magic, I use reason, and I am always quite conscious of the details of the reasoning. Except once . . ." He looked thoughtful.

Drake lit a cigarette from the dying stub of hid old one and said, 'Tell us, Simon."

"I intend to," said Alexander, "but it has a certain personal confidentiality about it. I have been given to understand that everything that goes on here is confidential."

"Everything!" said Trumbull pointedly, "and that includes everything we say. We take it for granted that nothing you hear here can be used against us as tar as our taxes are concerned."

"Agreed," said Alexander, "but please be careful what you say, as I would rather not be asked to place too great a load on my integrity."

Alexander sipped at his curacao, looked pensive, and, for some reason, particularly Satanic.

"You know, of course," he said, "that computers are now the lifeblood of t lie IRS. We couldn't operate without them. Because they never hesitate, never tire, never grow bored, they are our great strength. Because they never think, they are our great weakness.

"To exploit a computer, however, and take advantage of its weakness, we must know every detail of a computer's working, and this eliminates almost all the human race. And it puts us off our guard.

"Some years ago, the IRS was royally diddled by someone who knew his computers, by a mathematician who was tired of the type of remuneration that mathematicians receive."

Halsted, who taught mathematics at a junior high school, sighed and said, "I know the type."

"The long details of his operation don't concern you, but he managed to get a job that would involve the servicing of certain of our computers. For the purpose, he built himself a new statistical background, a new name, a new appearance, all the way down to a new Social Security number. How he managed that I won't tell you since, confidentiality or not, there is no point in spreading knowledge of the techniques of successful knavery."

"I agree," said Avalon, nodding.

"Nor," said Alexander, "will I tell you exactly how he managed to reprogram one key computer - for in that case, I don't understand it myself. I am no mathematician. Still it was done. For a period of five years, our mathematician - let us call him Johnson, to save syllables - received large tax refunds while paying no taxes. He received more money in that interval than he could have earned in a lifetime of honest endeavor.

"He might still be receiving the money but for the accidental uncovering of an inconsistency in the program. The detection was the result of a most unusual coincidence, and I assure you the IRS could not have been more dismayed, or embarrassed, at the event. Naturally, two things were at once essential. The money leak must be stopped and the computer programs so modified as to make the Johnson type of knavery impossible in the future. That was carried through at once in the greatest secrecy. The secrecy was needed not so much to keep individual officials from looking personally ridiculous, though that was a factor, but to keep the Service itself from losing the confidence of the American people."

"They will never learn the truth from any of us," said Gonzalo, with suspiciously intense gravity, "I assure you."

"The second thing," Alexander went on, "was to catch Johnson, make him disgorge what money was left, and clap him in jail for as long as ever the law would allow. It was the Al Capone reasoning, you see. Johnson might get away with murder without shaking the foundation of American civilization, but he could not be allowed to get away with income - tax fraud. And that was where I came in. I was placed on the case.

"My reputation in the Service is, perhaps, an exaggerated one. More than one person there suspects, as Jim told you, that I solve my cases with some inner magic, with some mysterious intuitive faculty that defies analysis. It has been said among us, for instance, that I can look at a tax return that seems clean enough to have been etched in snowflakes and yet tell that somewhere money was clinging to fingers that smelled of garbage. Or that I could interview a person and know for certain that there was a thief hidden behind the saint.

"Actually, there was no magic in it at all. I have a certain cleverness at observation and reasoning and a great deal of experience. My memory is excellent and I have encountered all varieties of behavior patterns - and all the ways of laundering returns, too. What seems like magic or intuition boils down to noting small things that others don't and attaching the proper importance to those things.

"It works the other way around, too. I can often detect the saint beneath the thief. I am quite certain, for instance, that you, Mr. Trumbull, are not short on your returns by even as much as fifty dollars. I suspect that you are ashamed of your relative honesty, and take it out by vilifying the office you dare not defraud. And that's not a guess; I've met others like you."

It was hard for Trumbull to flush through his tan, but his expression made the flush unnecessary.

Avalon said, "I'm afraid your reputation is ruined, Tom. Please go on, Mr. Alexander."

"To put it in figures, in nearly a quarter of a century at my job, I have almost never pointed a finger in the direction of either guilt or innocence and been proven entirely wrong."

"A quarter of a century?" said Avalon. "How old are you, Mr. Alexander?"

"Fifty - two."

"You don't look it," and Avalon's finger went unconsciously to his own graying mustache.

Alexander said, "There's gray in my hair, too, but I touch it up a little. Not so much out of vanity, you understand, as because the darkness seems to give me a forbidding appearance that is useful to me in my line of work. However ..."

"Johnson was not an easy quarry. He could tell, somehow, that the game was up, and when the next refund came - this time under the eyes of the Service - it remained uncollected. It's not impossible he had an ally within the Service, but never mind that. Tracing him wasn't easy. He had quit his job long before and all records we had concerning him were false, down to his Social Security number, which, we suddenly discovered, was attached to no human being.

"I was forced to follow the most evanescent clues and to build up the picture of the human being who had done the deed. We left absolutely nothing unturned that might lead us to the identity of the thief and we finally had several possibilities, all dim and uncertain. Different operatives were assigned to each one. The task was somehow to locate enough evidence to warrant a concentration of forces, a full - scale investigation of one particular man - an arrest, of course, if possible.

"My own target was a rather mousey man of average weight and height and of undistinguished appearance. That, in itself, was a good sign, because the job had had to be carried through by someone who could be unnoticeable at crucial moments. He had a vague background that could not easily be traced without tipping our hand too soon - again a hopeful circumstance. At crucial periods, he seemed to be particularly untraceable.

"Unfortunately, all this was negative and such things could never be made to stick. We needed some positive correspondence. We had to locate him at the site of action each time, prove computer expertise, and so on. For that, I haunted him like the ghost of a vulture.

"In fact, I managed to collect a circle of mild acquaintances that I held in common with him, and I labored to be present at social gatherings along with him.

"Then, at one gathering in early November, where both of us were present, he quiet and watchful, nursing a single drink for an hour; and I almost as quiet, certainly as watchful, and as abstemious, the host spoke of Halloween. He had a seven - year - old daughter who had gone out trick - or - treating, along with several older friends, and who had come back in ecstasies.

"That rang true for me, for I remembered very well my own daughter's first experience of the sort and I said, 'Yes, I have always thought that, were it not for the enormous commercial overweighting of the Yuletide, a child's spontaneous reaction would be to treat Halloween with the full excitement of Christmas.'

"And, surprisingly, my quarry spoke up. As though overwhelmed by an emotion that forced his naturally quiet personality into the limelight, he said, with a warm smile that lightened and almost transfigured his face, 'You are quite right. In a way, Halloween may be considered precisely equal to Christmas.' Those were his exact words, gentlemen, for I noted them at the time with particular care."

Rubin asked, "Why?"

"Because, as a result of that comment, I instantly and completely eliminated him as a suspect. So certain was I that I remember having the distinct impulse to clap him on the shoulder and invite him out for a drink to celebrate his innocence. I couldn't, though, for I suppose his own unexpected warmth had scared him. As soon as he made the remark, he flushed, looked frightened, and melted away. My own attention was distracted for a moment, and when I turned to find him, he was gone."

Alexander paused and finished his last bit of curacao. He said, "At the time, my sudden conviction of innocence might have seemed pure intuition - even to myself - but it wasn't, of course. I cling to Rubin's hypothesis the intuition is undetected reasoning. Here is the reasoning as I worked it out later.

"Working laboriously from the tiniest beginnings, we had drawn a picture of our criminal, this Johnson. He was a mathematician and had no family. The chances were that he was not only unmarried and childless, but also that he had no siblings and that his parents may have died while he was young. He was cold, utterly cold - and I don't mean by that that he was ruthless and sadistic - merely that he lacked any occasion or desire for love and affection. Let me put it in a way that has great significance for me. He was not, in any way, a family man."

Halsted pleated the tablecloth absently and said, "You, I take it, are a family man."

"Completely. My parents, two brothers, and a sister all live, and we are all close. I married a childhood sweetheart, have three children, and a grandson newly born, plus nieces and nephews. I know the emotions of a family man and no one, no one, could have spoken with such genuine warmth of children's holidays unless he had experienced the kind of love and affection that accompanies those days. My quarry spoke that way; Johnson could not have; conclusion, my quarry was not Johnson and was innocent. What seemed like intuition was, after all, reason.

"Intuition or reason, I reported my belief in his innocence to my superiors, and the tracking of the remaining possibilities grew correspondingly more intense. Five months later we caught the criminal, and he is now in prison and likely to remain there a long time. Some of the money has been recovered; not all, of course."

He paused and Avalon broke the short silence that followed, saying, "I am delighted to have a happy ending for the department, but you spoke of a puzzle and I see none."

Simon Alexander sighed, "The happy ending is a qualified one. Having dismissed my suspect, we nevertheless found the other suspects fading as well. One after another, they proved to be incompatible with the conditions of the fraud. One day, out of nothing but desperation, I returned to my own dismissed suspect and, in the interval, something unexpected that had arisen cast a new light on affairs. Astounded, I followed it up and had him - my own suspect whose innocence I had previously maintained and, virtually, guaranteed. He was the criminal after all.

"What puzzles me and, even now, keeps me awake occasionally, is the incongruity of it. He did indeed turn out to be what we had suspected he was - a man without family, love, or affection. Yet his remark about Christmas and Halloween, and the tone in which it was uttered, indicated the reverse. How is this contradiction possible and how could he have used it to throw me off the scent?"

There was a silence around the table as Alexander waited for an answer.

Avalon finally spoke, staring at his empty brandy glass, "Mr. Alexander, despite easy theorization, the fact remains that human beings are complicated and inconsistent creatures. There are undoubtedly contradictory aspects in the character of your suspect or of any man. You'll have to chalk it up to a bad break."

"I'd like to," said Alexander, "and I've tried to do so, but it is my experience that in fundamentals human beings are not inconsistent. A man who always puts on his left shoe first, may switch party allegiance and swap wives, but he will always put on his left shoe first."

"Nothing will stop him from putting on his right shoe first, however uncomfortable that might be," said Halsted, "if it is necessary to do so to fool someone. He stepped purposely out of character to mislead you."

Alexander didn't answer at once. Then he said, "I doubt that. Even if he knew I was at his heels, and that's a possibility, he couldn't possibly have known me so well as to be sure that one short, apparently irrelevant sentence would deflect me."

Rubin said, "The remark might have been misanthropically in character, and you interpreted it wrongly because of your own happy associations with the holidays. What the suspect might have meant was that Christmas was just as superstitious and nonsensical as Halloween."

Alexander said, "An interesting thought, but the expression on his face and the tone of his voice did not fit. They were happy, delighted. I am still sure he meant it sincerely as a loving remark."

Gonzalo said, "He could be a 'Peanuts' fan and was thinking of Linus's 'Great Pumpkin,' which is a kind of satire on Santa Claus. That would set up a strong association between Halloween and Christmas."

There was a general hoot from the audience, but Alexander held up his hand. "Actually, that's the first suggestion I hadn't thought of myself. It doesn't sound in the least likely to me, but I will check whether the thief was a 'Peanuts' fan."

Trumbull said, "We don't have enough to go on. I don't think anything can be deduced from what he said to set your mind at ease. Sorry!"

Drake said, "I agree, but we haven't heard from Henry yet."

"Henry?" said Alexander in surprise, swiveling in his seat.

Henry cleared his throat. "I admit, gentlemen, that a thought had entered my mind at the moment Mr. Alexander gave us the suspect's remark."

"Oh?" said Alexander, "and just what thought was that?"

"The suspect, sir, did not say, as you or I might have said, that Halloween was just like Christmas or just as good as Christmas, or even equivalent to Christmas. If you quoted him correctly he said Halloween was 'precisely equal' to Christmas. Surely that sounds like a mathematician speaking and would be in character."

Alexander snorted. "Feeble. Feeble. A nonmathematician might have happened to put it that way if he were a prissy and meticulous person."

Henry said softly, "Perhaps. Yet we might find more in the statement if we treat it mathematically than anyone has yet pointed out. After all, it your suspect were indeed the guilty man he would be not only a mathematician, but also a computer specialist."

Alexander looked annoyed. "What has that to do with it?"

Henry said, "Mr. Alexander, listening to the gentlemen of the Black Widowers month after month is an education in itself and there have been times when I have directed my readings in the directions they have opened up for me. Mr. Halsted, for instance, once discussed the rationale behind positional notation, the manner in which our Arabic numerals are constructed, and I went on to read further concerning the matter. If you'd care to have me explain it, I'm sure Mr. Halsted will be glad to correct me if I've made a mistake."

Halsted said, "I'll be glad to, Henry, but I don't see what you're driving at."

"You will in a moment, sir. Our ordinary numbers are written to the base ten. The first column at the right are the ones. The next to the left are the tens, the next are the hundreds or ten times tens, the next are the thousands or ten times ten times ten, and so on. Thus the number 1231 is one times a thousand plus two times a hundred plus three times ten plus one and that comes to one thousand, two hundred thirty - one."

"Right so far," said Halsted.

"But there's no need to consider ten the only possible base for a number system," said Henry. "You could use nine, for instance. The right - hand column in a nine - based system would be ones, the next to the left would be nines, the next would be eighty - ones or nine times nine, the next would be - uh - seven hundred twenty - nines or nine times nine times nine, and so on. The number 1231 would, in the nine - based system, be one times seven hundred twenty - nine plus two times eighty - one, plus three times nine, plus one. That would be, if you would allow me a moment to work it out - the equivalent of nine hundred nineteen in our ordinary ten - based system."

He scribbled hastily on a napkin and held it up. "You could write the result this way: 1231 (nine - based) ==919 (ten - based)."

Halsted, Drake, and Rubin nodded. Avalon and Trumbull looked thoughtful, and Alexander shook his head impatiently.

Gonzalo said, "That's ridiculous. Why would anyone use that nine - based system and multiply nines?"

Halsted said, "The other number bases look complicated, Mario, only because our number system is designed to fit ten as a base. Mathematically, all are equivalently rational, though some are more convenient than others. For instance, in computers, it is particularly useful at times to use - uh, oh . . ."

He looked at Henry with a grin and said, "I may be getting it. Henry, but you keep on and finish."

"Thank you, sir," said Henry. "As Mr. Halsted was about to say, the eight - base system is, I understand, useful to computers. The number 31, for instance, in the ten - base system is, of course, three times ten plus one, or thirty - one. In the eight - base system, however, it is three times eight plus one, or twenty - five.

"We can therefore write" - he used the napkin again - "this: 31 (eight - base)=25 (ten - base).

He went on, "The different number bases are sometimes given names derived from the Latin names of the numbers. The Latin for ten is decem, so a ten - base number belongs to the decimal system. The Latin for eight is octo, so eight - based numbers are octal. We can therefore write this: 31 (octal)=25 (decimal).

"By coincidence, we have the months October and December. . . ."

Rubin roared out in sudden delight, "No coincidence at all. The ancient Romans started their year in March before Julius Caesar's time. By that system, October was the eighth month and December the tenth month and they were named accordingly."

Henry nodded his head and said, "Thank you, sir. If, then, we abbreviate the terms 'octal' and 'decimal' in a natural way and omit the parentheses, we have 31 October=25 December. How can this be described better than by saying that Halloween, which falls on 31 October, is precisely equal to Christmas, which falls on 25 December."

Alexander's mouth had tended to slacken through this but now he tightened his jaw muscle and said, "Are you trying to tell me that the thief, blurting out his remark without thinking, gave away the fact that he was a computer expert?"

"Yes, sir," said Henry. "It was not out of shyness that a look of alarm crossed his face and that he left as soon as he could. It must have been out of fright at the thought of having slipped into his real character. At that moment, if the significance of his remark had been seen by you, it would have been wise to begin arrangements for having him arrested."

Alexander looked chagrined. "Well, I didn't see it. I interpreted it just wrong. But wait, all this is clever and may even be right considering that the man in question proved to be indeed the criminal, but how would you account for that look of love on his face? That was what threw me."

Henry said softly, "You are a family man, sir, and that is your weakness. You naturally interpret love in human terms alone. I, myself, am not a family man, and I know that love is broader than that. Even a misanthrope who hates the human race could love, and deeply, too."

"Love what?" said Alexander impatiently.

"The beauty and surprises of mathematics, for one thing," said Henry.

"The Family Man' - Afterword

Those of you who have read the first two volumes of the Black Widowers tales know that there is a real organization called "the Trap Door Spiders" to which I belong, and which serves as an inspiration for everything about the Black Widowers, but the mysteries. Occasionally, I even make use of a real - life guest - at least for appearance and general background.

For instance, there was indeed a meeting of the Trap Door Spiders at which a magician, very much like the Amazing Lanni of the first story in this volume, did dine with us as our guest. (There was no mystery introduced, of course, in the real - life occasion.)

And at another meeting, which I hosted, I brought my accountant as my guest. He is a very amiable gentleman with whom I share a million laughs as he strips me to the bone for the sake of good old Uncle Sam and then gnaws at the bones for his fee.

I used him as an inspiration for Simon Alexander in "The Family Man," changing him from an accountant to a tax auditor (the enemy) and carrying on from there.