Disclaimer: Joss Whedon created the Angel character and Isaac Asimov created The Black Widowers.
X X X X X
And the Black Widowers are . . .
Geoffrey Avalon, patent attorney; James Drake, chemist; Mario Gonzalo, artist; Roger Halsted, junior high math teacher; Manny Rubin, mystery novelist; Thomas Trumbull, US government code expert.
Henry Jackson, waiter.
All are considered members. Especially Henry.
X X X X X
This particular meeting of the Black Widowers had gotten off to a rancorous start, and for once, the origin had nothing to do with a problem between Mario Gonzalo and Manny Rubin.
As Thomas Trumbull came bouncing up the steps, he said, “Scotch and soda for a . . . what’s going on here?”
The imperturbable Henry was there immediately by his side, with the scotch and soda Trumbull had never finished ordering. “It concerns Mr. Halsted’s guest,” he said.
“What about him?”
“That would be the issue, sir,” Henry said. “Mr. Halsted’s guest is not a him.”
“That’s against the bylaws,” Trumbull said. The Black Widowers individually were far from misogynist, but it was a rule only broken once in recent memory that the members were strictly male, and that all guests had to be men as well.
“Which would, I believe, be why they are having the debate,” Henry said. “If you’ll pardon me, sir, I need to check on the status of tonight’s meal.”
“Of course,” Trumbull said, and moved closer to the center of the argument.
While Rubin and Gonzalo had not begun the argument, though, they were at the center of it.
“No!” Gonzalo said. “A hundred times, a thousand times, no!”
“We let it happen once before,” Rubin said.
“Yes,” Geoffrey Avalon said, raising himself up to his full height of seventy-four inches, “But those were exceptional circumstances.”
“I’m inclined to agree,” Drake said.
“These circumstances are exceptional as well,” Halsted insisted. “Look. Would it help if I told you she was nowhere near here?”
“Has she developed a working transporter?” Gonzalo asked acidly.
“No need,” Halstead said, and reached into his pocket and pulled out a cellular phone.
Gonzalo threw up his hands. “First he wants to bring a woman here, now he wants to bring in cell phones.”
“There’s nothing in the bylaws about them,” Trumbull said mildly.
Everyone turned to look. “Sorry, Tom,” Rubin said. “We didn’t even know you were here.”
Drake said, “No, there isn’t anything in the bylaws about cell phones. That’s because we really didn’t think anyone would bring one.”
“It is turned off, I hope,” Avalon said.
Halsted said, “It is, and will remain so until we’ve hashed it out. But my ‘guest’ is perfectly willing to communicate with us solely by means of this phone. You don’t even need to see her face.” After a second, when no one answered, he added, “Would it help if I told you that her name was Fred Burkle?”
“A girl named Fred?” Trumbull asked.
“Winifred,” Halsted said.
Gonzalo said, “Enough. Now that Tom’s here, I call for a vote.”
“Seconded,” Rubin said.
“The proposition entails allowing Miss Burkle to talk with us via phone after we’re done our meal?”
Halsted immediately said, “Aye.” A few seconds later came Gonzalo’s nay, and after that, Drake’s. Rubin voted aye, and Avalon said, “Nay.”
That left Trumbull, who, after thinking for a moment, said, “Aye.” At Gonzalo’s groan, Trumbull added, “Twice in -- how many years is it? -- doesn’t seem like too many occasions on which to waive the rules. Every time, I’d be opposed to. Not this.”
“It doesn’t matter,” Halsted said resignedly. “It’s a tie, which means the motion fails.”
“It’s not a tie,” Rubin said. “It can’t be a tie. Our seventh member hasn’t voted yet. Henry?”
Henry entered the room. “Here, sir.”
“You know what we’ve been discussing, right?”
Henry smiled slightly. “I could scarcely avoid the knowledge. It was a rather heated conversation.”
“And what do you think?” Gonzalo asked.
“My opinion would be that if the young woman never sets foot in the establishment -- that if she is, in fact, potentially hundreds or thousands of miles away --”
“About ten, actually,” Halsted said. “She’s been attending the same mathematician’s conference that I have.”
“Ten, then,” Henry said. “Still, if I may, she is not here. Therefore I see no violation of the bylaws. My vote is also ‘aye.’”
Halsted grinned in relief. “That’s settled, then. Good.”
“More importantly,” Henry said. “Gentlemen: Dinner is served.”
Dinner itself was subdued, even though the food was easily up to the Milano’s usual standards, opening with a nice chicken-and-mushroom soup followed by a grilled swordfish steak and vegetables and finished with an apple tart. What conversation there was was somewhat strained, though everyone made a game attempt to get past the earlier debate.
Once the plates had been cleared and everyone was sipping their coffee and brandy, Halsted said, “The first thing I need to do is describe who our guest is, since we haven’t had the chance to talk to her and I don’t want you coming in cold. I met Winifred Burkle earlier this week at the Columbia Mathematics conference; she was asking for help on a problem she was having. Assorted mathematicians from around the world attempted to come up with answers, and had no luck.”
Drake said, “And you think that we will be able to solve a mathematics problem that professional mathematicians have been incapable of solving? I think you overestimate our abilities, Roger.”
“If it were strictly a mathematical problem, I would agree with you. I would no sooner expect any of you to be able to figure out a flaw in an alleged proof of Fermat’s Last Theorem – and yes, there are still people trying to come up with a more elegant solution than the one that has already been promulgated – than I would expect you to suddenly leap from the window and fly around the building. But it’s not, apparently, strictly mathematical.”
“Is Miss Burkle a mathematician herself?” Rubin asked.
“No,” Halsted said. “She’s a physicist. But one with an impressive grasp and knowledge of mathematical history. Not quite a polymath –“
“I fear we are beyond the age of the true polymath,” Avalon said. “The sciences are so specialized these days that acquiring a basic knowledge of each seems about as much as any reasonably intelligent person can muster.”
“As far as I’m concerned the only true polymath was Leonardo da Vinci,” Gonzalo said.
“We might disagree there, Mario,” Halsted said. “But, while she’s not a polymath, she has a fairly broad knowledge of most fields of science, as near as I can tell.”
“What is the problem?” Trumbull asked.
Halsted shook his head. “I prefer to let her tell you. I’ve already spent some time thinking about this and I want you to come at with a fresh set of eyes.” After a second, he said, “Also, can we dispense with the opening question?”
Everyone agreed, some reluctantly. “Good. Manny, if you’d take the lead?”
“Sure,” Rubin said.
“Good,” Halsted said, “Are we ready?” then, over to one side of the room, “Henry?”
“I’d like you to be able to hear this as well. The phone’s speaker is fairly loud, but I’m not sure how well it will carry to the furthest corners of the room.”
Henry moved until he was standing just slightly behind between Trumbull and Gonzalo. “Will this do, sir?”
“That’s fine,” Halsted said. “Now . . .” and he picked up the cellular phone and dialed the number. “Hello, Fred? This is Roger Halsted. They agreed to help.”
“Hold on,” Gonzalo said insistently.
“Hold a second,” Halsted said, and pressed a button. “Yes?” he asked Gonzalo.
“You told her about us?”
Halsted said, “Don’t be an ass, Mario. I know it can be difficult at times. All I told Miss Burkle was that I knew of a group of people who specialized in solving problems who might be able to assist.”
“Seems nonspecific enough to me,” Drake said.
Gonzalo nodded. “Just be careful.”
“I always am,” Halsted said, before returning to the phone. “Fred? Sorry about that. I’ll place you on speaker.”
Then he hit another button and placed the phone down as close to the center of the table as he could reach. “Fred? We can all hear you now.”
“Hello, everybody,” a mild and somewhat breathless said from the phone. It had a distinct Texas accent. “I’m Fred Burkle.”
The Black Widowers went around the room, briefly introducing themselves, finishing with Henry.
“Nice to meet y’all,” she said sweetly. “Has Roger told you anything about my situation because if he has it’d be easier for me just to jump right in and ask if you have any questions.”
“Actually,” Rubin said, “All he said was that you had a problem. He preferred to let you explain it. Would you do so?”
“Okay, I’ll do my best.” She audibly took a deep breath. “I run the science division of a large company out in California.”
“What kind of company?” Rubin asked.
“A law firm, and before you ask I know it’s not usual for a law firm to have a science division but this isn’t your usual law firm and we don’t deal with your usual clients. We deal with unusual circumstances and unusual objects. One of them’s what’s causing the problem right now. One of our clients has some very valuable – um, artifacts –“
Gonzalo jumped on the hesitation. “What kind of artifacts?”
“One of the conditions,” Halsted explained, “Is that you have to answer any question. Anything you tell us will remain confidential.”
“Yes,” Miss Burkle said, “But you know cell phones aren’t as secure as the manufacturers like to pretend and even though we made the one on our end ourselves I don’t think the security applies to the one on your end. Don’t worry, I’m not going to lie to you about anything ‘cause I need the help too much to do that, I’m just going to have to make things a bit vague. Sorry.”
“Something on the order of valuable art, then?” Avalon said, trying to offer a compromise.
“Yes,” Miss Burkle said. “But art that can be used for something else.”
“Like a piece of fine china?” Trumbull asked.
“More important than that, something more like Raiders of the Lost Ark. Though without the boulders and the Nazis. And of course I’m only talking in generalities here because the specifics really aren’t important. The problem is, these artifacts are in a kind of safe.”
“Is it your safe?” Rubin said.
“No -- it belongs to one of our clients -- Victor Baines” Ms Burkle said. “And the reason we’re not asking him is at the moment he’s kind of dead.”
“Kind of?” Gonzalo said.
“Okay, really he’s all the way dead, but anyway he can’t answer our questions about how to get into the safe, and he never told anyone the combination. I suppose I should tell you also that it’s not like a typical safe; it’s got two locks, and each has a keypad in front of it and a little screen that can hold up to fifteen individual digits each -- that’s a quadrillion times a quadrillion if we tried it randomly.”
“No point to that, then,” Avalon said.
Trumbull said, “I imagine there’s some kind of security device built in as well, to disable the lock if too many wrong codes are entered.”
“Exactly,” she said. “So we want to make sure we get it right.”
“So,” Drake said, “Since you asked a group of mathematicians, I’m guessing you have some kind of mathematical clue.”
“I do indeed, Mr. Drake,” Ms Burkle said. “He told his assistant that ‘the first and last number’ would get him into the safe if he ever needed to get there. The assistant doesn’t have normal access to the safe so she never got to try it out, and anyway she wasn’t able to figure out what Mr. Baines meant -- she‘s not a mathematician herself -- and we aren‘t able to either.”
“The first and last number?” Rubin asked. “There is no such thing.”
“And yet that’s all we have to work with,” she said.
“How sure are you that the assistant is reporting what Baines said accurately?” Avalon said.
Ms. Burkle’s response was, “Pretty sure, after all it is just three words, it’s not like she had to memorize a paragraph.”
“Hmmm. I was wondering if maybe it was a Biblical reference. The beginning of the book of Revelation says,” and his deep voice practically boomed, “I am Alpha and Omega, the first and the last.”
“Good idea, Mr. Avalon,” Ms. Burkle’s voice came. “Unfortunately, while omega has a mathematical meaning -- it’s the smallest infinite ordinal -- it can’t really be identified with any particular number. And alpha doesn’t mean any number at all.”
“It’s still an obvious reference,” Rubin said. “What verse was that?”
Henry said, “Chapter 1, verse 11.” He was holding a copy of the King James version that he had unobtrusively plucked from the Black Widowers’ reference shelf, which also contained two unabridged dictionaries, an almanac, and a current Manhattan phone directory, among other things.
“Thank you, Henry,” Rubin said.
“The problem is,” Drake said, “That 1:11 is not in itself the first and the last; it’s simply where we quote it from.”
Ms. Burkle said, “I’ll go along with that. Anyone else?”
“Could it have simply been that the first and last number that were capable of being entered into the system?” Gonzalo asked. “A zero on the left-hand side, and fifteen nines on the other?”
“We actually already tried that,” she said. “It didn’t work.”
“Are there any provisions for letters or decimal points or any other symbols on these keypads?” Trumbull asked.
“No. Not even two extra keys to balance it out like they have on ATM’s and pushbutton phones, and no letters on the numbers, either.”
“At least there’s that,” Gonzalo said.
“It’s a minor check on the positive side, which I suppose I should be grateful for,” she said.
“Was Mr. Baines, himself, a mathematician?” Avalon said.
“No, but he was a fairly educated man. He wouldn’t have known a quaternion if he’d tripped over one, but he didn’t have to take off his shoes and socks to count past ten.”
“What was he, then?”
“A businessman,” Ms. Burkle said.
Is he a sports fan?” Rubin asked.
“Not sure if he is or not. Why?”
“I was thinking that the numbers on athletes’ uniforms tend to be 0 at lowest, and 99 at the highest,” he said.
“They often use 00 in basketball,” Drake said.
“I don’t think you’re on the right track here, Manny,” Halsted said, breaking his silence. “00 and 99 would be the highest and lowest numbers, not the first and last.”
“Isn’t it the same thing?”
“Not really,” Ms. Burkle said. “They can be the same thing but it’s more often than not just a coincidence. And if we’re thinking like that in other realms I suppose somewhere there’d be someone who was the first to have a number on their jersey or whatever but there can’t be a last one. Since people are still playing and all,” she added apologetically.
After a moment of silence, Avalon said, “Perhaps we could approach this from another angle. The first number could very easily simply be 1.”
“Actually, it sounds to me like zero would be acceptable,” Drake said. “Am I right, Miss Burkle?”
“You are, Mr. Drake. But not multiple zeroes. Even though we’ve only entered the one combination we have tried fiddling with the keypads and it’s not going to take any initial zeroes. Which is good because that would add about a quadrillion more numbers into the system, not that we’re going to solve this randomly in any event.”
“And if the first number is one, Geoff, the last number is . . .?” Trumbull prompted.
“I know!” Gonzalo said, snapping his fingers. “Infinity.”
“Infinity doesn’t really count as a number, Mr. Gonzalo; it’s simply a concept,” she said.
“But would Mr. Baines have known that?” Avalon persisted. “There is certainly a common perception of infinity as being simply an extremely large number. This is something that a group of mathematicians wouldn’t know.”
“It’s all academic, anyway,” Drake said. “There’s no key for infinity on any keyboard I’ve ever seen.”
“Not even the most advanced scientific calculators,” Halsted confirmed.
“There are some advanced tools that let you fiddle around with it but there isn’t one here,” Ms. Burkle said.
Gonzalo wasn’t done yet. “But there is a number eight, right?”
“What are you talking about?” Rubin asked.
“Turn an eight on its side and it becomes a reasonable copy of the infinity symbol,” was Gonzalo’s response. “Zero on the keypad on the left, eight on the keypad on the right.” At everyone’s hesitation, he continued, “If you’re concerned that Baines might not have known what the infinity symbol stood for –“
“That’s not it, Mr. Gonzalo,” Ms. Burkle said. “I don’t know whether he would have or not – it’s not uncommon but it’s not common either. The problem is that even if an eight on its side is infinity, an eight with normal orientation is just that. An eight.” After a second, she added, “Still, it’s an original idea, and probably the best one so far – no offense to everyone else, of course.”
“You mean, none of the mathematicians you asked --?” Rubin asked.
“No, no, they came up with it and dismissed it in the same breath. I guess to them the eight-turned-on-its-side thing seemed kind of childish,” she said almost apologetically.
Manny’s guffaw interrupted her. “Figures you’d come up with the childish answer, Mario,” he said, laughing.
“And yet it’s still the best thing Ms. Burkle’s heard so far,” Gonzalo said.
“Unfortunately,” Avalon said, “I suspect that ‘best thing she’s heard so far’ isn’t going to be good enough. The problem is that the clue is far too vague.”
“And yet, the thing is, Mr. Baines was apparently convinced it was good enough for his assistant to be able to figure it out, if he had to. Any other suggestions?” After a brief silence, she said, “Well, then, thanks, and I guess we’ll try the zero and eight --”
“Pardon me, Miss Burkle?”
“Yes? Mr. -- Jackson, right?”
“Henry, miss,” he said. “If I may?”
“Of course,” she said.
“I would like to suggest that you take Mr. Gonzalo’s suggestion and reverse the numbers.”
“Are you sure?” she said.
“Quite sure, miss,” he said. “You said that Mr. Baines would not have given the clue if he hadn’t thought his assistant, whom you stipulated was not a mathematician, could figure it out. That would seem to me to rule out anything overly complicated. But Mr. Avalon is also correct that it is seemingly very vague. Therefore it must be something that is so obvious that, once explained, it will be more apparently correct than simply sporting uniforms.”
“How is your suggestion obvious, Henry?” Halsted asked.
“It had occurred to me that perhaps the answer was not mathematical at all. We are accustomed, of course, to thinking of ‘number’ as a mathematical concept -- but what we overlook is that numbers are also words. And in English, eight is first and zero is last alphabetically. And this is true even should we extend it to infinity among the integers, in both directions.”
“Billion?” Drake said.
“Properly, the number is one billion,” Henry said. “Does this seem logical to you, miss?” After a second, “Miss?”
A delighted voice came from the other end of the phone, “It more than seems logical, it worked! I called my boss on the other line and he tried it and boom! The safe’s open and we have the Ankh -- I mean, we have what we need. Thanks, Henry!”
“You’re quite welcome, miss,” he said.
“And thanks to all of you for helping out.”
“All of who?” Gonzalo said. “It was Henry who figured it out.”
“You did your parts,” Miss Burkle insisted.
“Quite so,” Henry said.
X X X X X
This doesn’t directly tie in to any fifth-season episode, of course; take it that the backstory, which the Black Widowers didn’t know about, involved the necessity of opening that safe because there was a specific artifact kept inside that was necessary to complete some important ritual or other, and that magical means of getting it open weren’t an option because the safe was appropriately booby-trapped. It’s not a story I’m ever going to write.