Azarya, away from his watchful older sister, was now able to indulge in his curiosity about these visitors, especially the lady whom he thought as beautiful as Queen Esther.

His first question to them was where they came from, fascinated to hear that they came neither from New Walden nor from Brooklyn nor from Eretz Yisroel, the Land of Israel. He could recount for them, and did, the seven generations of Valdener Rebbes and their wives and children, going all the way back to Reb Azarya ben Yisroel, who had been a direct descendant of the Besht. He knew exactly where he was situated on the family tree. But he didn't know that the name of the country he lived in was the United States of America. Roz wanted to draw him a map of America. He'd never seen a map of anything, and once she explained the idea of a map to him, he grew so excited that he went running out of the room, his silky blond side curls flying, to go find something to draw with. He came back a few minutes later with a box of crayons and a few sheets of coarse white paper.

Roz got down on the floor, since there was no writing surface in the room, with the little boy stooping down near her so that he could watch closely, his hands clasped between his knees. She drew a reasonably well-proportioned and accurate map of the United States, using red and blue crayons. She also drew an American flag for Azarya and explained about its stars and stripes. She colored the Atlantic and Pacific oceans for him. He knew about the ocean because of the splitting of the Red Sea. He had no idea that he lived within fifty miles of an ocean. One of his older sisters had been to Brooklyn, but he had never left New Walden. He hadn't realized that when his father had gone to Eretz Yisroel he had had to cross the blue water that Roz drew for him.

Azarya knew how to read Hebrew and Yiddish and Aramaic, but he hadn't been taught the English alphabet yet. Nevertheless, Roz labeled all the states, saying the names aloud as she wrote, and labeled "New Walden" and "Cambridge."

"This is where you live, and this is where Cass and I live. Maybe someday you'll come and visit us. Would you like that?"

"With my sisters, too?"

"Sure, why not." As long as they were dreaming the impossible anyway, they might as well make it to the child's specifications.

"Are you married?"

Azarya had settled down cross-legged on the floor next to Roz. He was as comfortable with strangers as the Onuma brats, which was remarkable, given the insularity of the Valdeners. Being the Rebbe's son, and a prodigy to boot, he'd probably been bathed in affection and powdered in praise his whole life.

"To each other? No." She smiled down at him.

"To someone else you're married?"

"No. We're not married at all."

"Will you invite me to the hasana?"

"The wedding," Cass explained to her.

"Of course we'll invite you! Do you think we'd have our hasana without you?"

The child broke into his wonderful smile.

"Now I'll draw you a picture!" He took another of the sheets and lay down on his stomach on the floor beside Roz with his box of crayons and got to work.

"It's a surprise for you," he told them. "I'm going to make one for each of you. For your hasana. Don't look yet."

He didn't get the chance, though, since the rail-thin woman who had taken Azarya and Roz into the Rebbe's study soon came to fetch Azarya. She, too, had some words of gentle chastisement for him, apparently having to do with his being on the floor. She glared at Roz, who was sitting there beside him. Azarya got up quickly and then handed the sheet of paper to Roz, telling her sadly that he hadn't had time to finish.

"The last one isn't finished. When you look at my drawing, you'll see many different maloychim," he said to her, for the first time looking a little bit shy. "I'm sorry," he said to Cass, "that I didn't make a drawing for you. Ble nadir, I'll make one for you next time." As he was being led away, he looked back over his shoulder and smiled the smile of a cherub, lifting his little hand and opening and closing his fingers to wave in the manner of the very young. Bye-bye.

"That's not only the most extraordinary child I've ever met," Roz said to Cass and Klapper as they walked back to the car in the quickly falling dusk, "I think that might be the most extraordinary person I've ever met."

Klapper stopped walking and looked at Roz.

"I think he must be some sort of idiot savant. The Rebbe spoke feelingly of the problem with genetic disorders that they have. It's the price of their marriage purity."

"Idiot savant! Are you kidding me, Jonas?" Cass winced at the many liberties Roz was taking. At least she didn't call him "the Klap" to his face-not yet. "If that kid's not a genius, then I'll eat his father's fur hat. Turn on the light, Cass." They were in the Lincoln Continental now. "I want to see that drawing he made for me."

Azarya had folded the sheet once in half. When she unfolded it, she saw a bunch of numbers written down, arranged in clumps that were somewhat triangular in shape, and that he'd written with different-colored crayons.

"Let's see," said Cass, and Roz held Azarya's picture up so that both Cass and Klapper could inspect it.

"Perhaps it's gematria," Klapper said, showing a momentary interest.

"What's gematria?" asked Roz.

"I'll explain it to you as soon as Reb Chaim turns off the interior lights and turns on the heat. When I say my teeth are chattering I am not speaking figuratively. Gematria, from the same Greek roots that give us 'geometry,' is an ancient anagogic means of extracting the hidden meanings out of sacred texts, by assigning numerical values to letters and then computing the values of a word or phrase. The Greek isopsephy, the Muslim khisab al jumal are examples of similar techniques, but the most intricately and cannily developed is the gematria of the Kabbalists. Take my name, for instance, which is a rather fascinating one."

" 'Klapper'?" Roz asked, and Cass felt a buzz of alarm. It would be catastrophic were the conversation to yield revelations of Klepfish.

"No, I refer to my given name, Jonas Elijah. In Hebrew, my name is Yonah Eliyahu, and if you take the gematria of my first name it adds up to thirty-five: yod is ten, vav is six, nun is fourteen, and heh is five. If you then add the first letter of my second name, aleph, you get thirty-six-or in Hebrew, lamed vav, which is a number with profound mystical significance. The Lamedvavniks are the thirty-six people of impregnable purity who live in every generation and for the sake of whom the world is not destroyed. Their identity is kept so secret that even they don't know who they are. The name Yonah means, literally, 'dove,' but also can mean, paradoxically, 'the Destroyer,' as it also means 'a Gift from God.' And in Hebrew the name of Yonah, yod vav nun heh, is almost identical to the one and only true name of God, yod heh vav heh, or Yahweh, the Divine Tetra-gram. The only letter that distinguishes between the two is the letter nun, which is a letter known as the 'winged messenger.'"

Boy, you never knew what Krap the Klap was going to throw at you. Roz almost had to admire him.

"Azarya's drawing had a bunch of zeros in it. Does the zero mean anything in gematria?" she asked him.

"Nothing at all. I'm afraid that the child's drawing has no Kabbalistic significance whatsoever. An indulged and, I've no doubt, much-tutored child has learned to write his numbers without having much grasp of what they mean and drew you a picture out of them."

"I bet there's more to it than that. That child is something wonderful."

"There was indeed something wonderful back there, young lady, and it is unfortunate for you that it passed you by. You are the sort who, should she witness the Messiah walking on water, would be impressed that his socks had not shrunk."

Cass marveled at how well Jonas Elijah Klapper kept his temper with Roz. He just hoped she wouldn't push it. His hope was in vain.

"I'm just wondering. Did you follow what he was talking about?"

Cass winced, waiting for the onslaught. But Jonas Elijah Klapper's response was astoundingly mild.

"Are you questioning whether I have the resources to keep pace with the Valdener Rebbe? Granted, he is a daringly speculative charismatic, but I hardly think we are mismatched."

"I meant did you follow Azarya."

"Azarya?"

"The Rebbe's little son."

"What I am failing to follow is you, young lady."

"I'm wondering whether you were impressed when he started talking about numbers."

"You think it had some Kabbalistic meaning?"

"I don't know about Kabbalistic. But the child had figured out-and, according to his father, all by himself-the concept of prime numbers. He'd figured out squares and cubes. He's six years old. That doesn't impress you?"

"No doubt he's an unusual child," Jonas Elijah Klapper conceded. His forbearance toward Roz verged on the miraculous. "He is, after all, of the royal line going back to the Ba'al Shem Tov himself. But, no, I'm not impressed by the slide-rule mentality. I remain unimpressed with the mathematical arts in general. What are the so-called exact sciences but the failure of metaphor and metonymy? I've always experienced mathematics as a personal affront. It is a form of torture for the imaginatively gifted, the very totalitarianism of thought, one line being made to march strictly in step behind the other, all leading inexorably to a single undeviating conclusion. A proof out of Euclid recalls to my mind nothing so much as the troops goose-stepping before the Supreme Dictator. I have always delighted in my mind's refusal to follow a single line of any mathematical explanation offered to me. Why should these exacting sciences exact anything from me? Or, as Dostoevsky's Underground Man shrewdly argues, 'Good God, what do I care about the laws of nature and arithmetic if, for one reason or another, I don't like these laws, including the "two times two is four?"' Dostoevsky spurned the hegemaniacal logic, and I can do no less."

They drove back to Boston in near silence, though Klapper had perked up a bit when they stopped at a McDonald's on the Massachusetts Turnpike, where he had expatiated on the parallels between the junk food on American highways and the junk ideas on American campuses, launching into the ludicrosity of English professors who study the evacuations from the posterior of popular culture-for example, a colleague he had had at Columbia who analyzed the lyrics of a musical ensemble called the Sex Pistols. Professor Klapper managed to expound even while he put away two Quarter Pounders with Cheese, a large order of fries, and a cup of cola the size of a cocktail shaker. "The fried potatoes are superlative," he said, his lips glistening.

They dropped Jonas Elijah Klapper off at his house in Cambridge, which was on an exclusive cul-de-sac pocketed behind the Episcopal Divinity School. The house was fronted by an iron gate and struck Cass as vaguely English. There was an old-fashioned lawn lamp shedding soft light on the ivy climbing the walls. He watched Jonas Elijah Klapper heavily mount the front stairs of his porch, unlock his front door, and disappear without a backward glance. He had sunk back into silence as soon as they had left McDonald's, and Cass wondered whether it was because of distaste for Roz, or because his visit with the Rebbe had yielded a feast of insights upon which he was meditatively chewing. Or perhaps he had been lulled into the tranquillity of an infant by the large car's effortless gliding through the darkness of night.

XIII.

The Argument from Taking Differences As tired as Roz had thought she was, she couldn't sleep. The first thing she had to think about, and dispose of, was just how infuriating that gasbag of a Klapper was. Did he actually say, without irony, "I've always experienced mathematics as a personal affront"? How could Cass not see him as one of the most prominent, if not the pre-eminent, propounders of poppycock of our day?

Cass was asleep beside her, his arms around her. He was the only man she'd ever known who liked to cuddle so much that he did it in his sleep. Sleep-cuddling was all that he had managed tonight. When she got out of the bathroom after brushing her teeth, he had already dropped off.

It was just as well. She'd felt put out with him. He'd given no indication that he opposed his demented adviser's ukase that Roz make herself as if she were not. Even the Valdener Rebbe had been less a misogynist than the Klap. In fact, she'd ended up liking him.

Poor boy. She predicted disenchantment of major proportions for Cass Seltzer. And if disenchantment never came, then its absence was an even more disturbing eventuality to contemplate.

But she didn't want to lie here and think maddening thoughts about the Klap. She disentwined herself from Cass's arms and crawled out of bed, going to the front closet to retrieve Azarya's drawing from her coat pocket. She crawled back into bed and switched on the little reading lamp. Cass stirred, looked over at her, smiled, and fell back asleep.

She studied Azarya's sheet, hoping it wasn't some sort of mystical gibberish that they'd given him in order to channel his interest in numbers into acceptable nonsense. Several minutes of study showed her it wasn't. She felt her scalp prickling, something that happened to her in moments of fear or excitement.

Azarya had color-coded his mathematics. In each pyramid, the line of zeros was in blue, the line after that, repeating a single number, was in green. The last line was in red, and the colors of the lines in between, if there were any, he'd left as they were, as if trying to show that they weren't as significant.

The first thing she noticed was that the last line of each of the triangles, the ones he'd written in red crayon, consisted of the first seven digits raised to a specific power. In the first triangle, he'd taken the numbers to the first power: 11, 21, 31. In the second triangle, his red line had the first seven numbers raised to the power of 2-12, 22, 32, 42, et cetera-which gave him 1, 4, 9, 16, 25, 36, and 49. Then he did the same thing for his third triangle, only now the bottom row had seven cubes: 13, 23, 43, and for the fourth triangle he'd raised the seven digits to the power of 4.

Roz got out of bed again and found her calculator. She used it to check the last numbers in his fourth last line. He'd gotten them right.

But, then, what about the lines above the red lines? It didn't take her long to see that the line immediately above the red line in each of the triangles was generated by taking the difference between the consecutive numbers in the line below. He did that, beginning at the bottom and proceeding up until he got to the green line, which had all the same numbers-odd that that kept happening-and which therefore gave him, when he took it to the next line above, his blue line of zeros. So his triangles were generated by taking differences. She went through with her calculator, checking his subtraction and her own surmise, and found both to be right.

But then came his fifth triangle, the one he hadn't finished. What was baffling was that he'd started not from the bottom but from the top. Why would he have done that?

The first line, the blue line of zeros, wasn't mysterious. Azarya had seen the pattern, the fact that taking the difference from consecutive numbers raised to the same power eventually gives you a line with all the same numbers, 1 when the power was 1, 2 when the power was 2, 6 when the power was 3, 24 when the power was 4. Azarya knew that the same thing was going to happen when he raised the first seven digits to the power of 5. He knew that, taking differences, he was once again going to get a line with all the same numbers, so that he'd end up with a line of zeros. So he'd written his blue line of zeros.

But what Roz couldn't see was how he knew that his green line would consist of 120s. How could he have known that before working his way up from the bottom, taking his differences?

What was the relation between 5 and 120, or, for that matter, between 4 and 24, and 3 and 6? What was special about 120? It was 10 times 12, which was 5 2 2 3 2. Or, in other words, 5 4 3 2. Or in other words, 5 4 3 2 1. It was 5 factorial, what mathematicians write as "5!" Roz's scalp was tingling like crazy. She only now noticed what was special about all the green lines.

24 equals 4!-1 2 3 4. 6 equals 3!-1 2 3. 2 equals 2!- 1 2. And 1 is equal to 1!-1 1. Azarya had drawn a picture for Roz showing the nth difference of xn is n!

One heard stories of this sort of thing, mostly in mathematics and music, the most self-enclosed of spheres. At five, Wolfgang Amadeus Mozart was composing ingeniously, if not yet immortally. It wasn't known until long after Gauss's death that the greater part of nineteenth- century mathematics had been anticipated by him before 1800, which was the year when he'd turned twenty-three. Generations of mathematicians had had to plod along behind him until they finally caught up with what he'd known in his adolescence.

And here was Azarya Sheiner cavorting with maloychim. No wonder the whole village cooed and petted over this child, rigid-faced women almost bursting into reckless laughter at the mere sight of him. He wasn't just the Rebbe's son, the Valdener Rebbetobe, the heir apparent, the Dauphin of New Walden. He was that accident of genes that happens only once in a very long while.

And even then, when such accidents happen, there has to be a blessed confluence of factors. A Mozart born to a family of slaves in the antebellum South would have created excellent spirituals while he was picking cotton in the fields, but not sublime operas. A Gauss growing up before the Arabs had invented the zero wouldn't have had a chance to carry mathematics into realms of infinite abstraction.

But when the alignment was right, then marvels ensued. If Azarya was discovering that the nth difference of xn is n! at six years old, then what would he be doing at sixteen, at twenty-six? The Rebbe himself had put it well. "There are children who are born as if knowing."

She wondered how much the child understood of what he'd drawn here. Had he discovered his theorem by playing with the numbers and noticing a pattern emerging, which would be astounding enough, especially since it would have required his discovering on his own the idea of raising numbers to a power-which he'd already spoken to them about- as well as factorials? Or had the child, even more astoundingly, discovered these patterns by seeing why they had to form? Did he see the reason for these patterns? Roz sure didn't. To her it just seemed uncanny, though not nearly as uncanny as a six-year-old's discovering it.

She looked over at Cass. He had put in a lot of driving miles, and under pressure, too, with that tyrannical buffoon breathing down his neck all day, oblivious to everything except his own obsessions-certainly oblivious to how emotionally complicated this trip back to New Walden must have been for Cass.

She gently swept the silky hair off of his high forehead, and the gesture of tenderness made her feel tender. Now she knew where that auburn hair came from. Redheadedness ran rampant in New Walden.

She shouldn't wake him. She herself would have snarled like a trodden cat if she was woken in the middle of the night because somebody with something to tell her couldn't control himself until the morning. But she couldn't control herself.

"Wake up, you crazy Valdener Hasid," she whispered in his ear. "Let me tell you about your next Rebbe."

XIV.

The Argument from Inconsolable Solitude Something jolts Cass into wakefulness at 2 a.m., and he can't get himself back to sleep. He had conked out early, falling into a deep and dreamless sleep, but now he's fully awake, groping around for provocations for feeling guilty, because that's the normal behavior when you wake up in the middle of the night. You wake, you feel guilty, you search for reasons to justify your guilt. Anyway, that's normal behavior for Cass.

It could be Shimmy Baumzer. Shimmy had served the salmon over guilt. The president had exacted a promise from him to think about "what sort of goodies you might like to see in a retention package.

"For example, I notice that you don't own your own house but rent a place in Cambridge. I could call up the Comptroller's Office right now, my friend," and he gestured with his elegant hand toward the phone, "and have them cut you a check that would cover the down payment for a house in Weedham. Maybe even in Cambridge. Another idea for you to kick over is whether you'd like to have your own Center for the Psychology of Religion, with a discretionary fund at your disposal. I can be creative, Cass. Just promise me you'll give it some thought."

Could it be around Lucinda that his unease is congealing? Yes, definitely. He's been holding back on her, not sharing the news about his offer from Harvard, and that's a troubling thought in the middle of the night.

His conversation with her tonight had been brief. She had been anxious to go through her PowerPoint one more time before getting her seven and a half hours of sleep. Rishi Chandrakar, the unworthy keynote speaker, had been mentioned again.

This isn't easy for her. She's such a proud person, in the best sense of the word. She had lifted up her transformed face to Cass in the twilight as he held the door of Katzenbaum open for her, and she had laid bare her vulnerability. She had been so terribly betrayed, both by the despicable David Prentiss Cuthbert, chairman of Princeton's Psychology Department, and by the system, and she's still bruised and uncertain, though nobody but he knows. Mona, for example, for whom his affection has cooled, hasn't a clue. Mona is very hard on Lucinda. Lucinda is right that she provokes irrational responses from envious people. It's obviously ludicrous to complain of being both brilliant and beautiful, and of course Lucinda isn't complaining, even when she sounds as if she is, but, still, she's been hurt by the people she calls griefers. His darling girl! He wishes he could be more helpful, but she's so beyond him. What can he say that will give her what she needs? He keeps trying.

"The first time I gave a talk at one of Pappa's conferences, he told me that if it had been any better he would have had to shoot me," she had told him tonight on the phone, sounding both proud and sad at the same time.

"Well, then, please don't make it any better this time," he'd responded, which at least had made her laugh.

The textbooks for his self-tutorial in game theory are piled up on his night table, and he decides to use his sleeplessness to make some more progress toward understanding the Mandelbaum Equilibrium. The farther he gets in the textbooks, the more he's been enjoying his foray into her science, finding himself increasingly resorting to its form of reasoning in order to clarify things for himself. The first thing to figure out always is whether a situation is a zero-sum game or not. Sum games are the ones where what's up for grabs-say, some pot of money-stays constant, and zero-sum games are the kind in which one person's gain is another person's loss: the addition of all the players' winnings add up to zero. You win, I lose; I win, you lose.

All sorts of situations can be analyzed as games, whether zero-sum or not. Take love, for example. Let's say you've got two people in a romantic relationship and neither has said "I love you."

Let's call them X and Y.

No, let's call them Cass and Lucinda.

What are the risks and what are the possible benefits of one of them saying "I love you" first?

Cass grabs a pen from Lucinda's night table, and, using the inside of the cover of one of the texts as his sketch pad, he draws himself some boxes: If Cass were to say "I love you" and Lucinda responded "I love you," which is what the top box on the left represents, then there would be a huge payoff. For Cass there would be bliss, and, presumably, for Lucinda there would be bliss as well, so the result would be bliss times two.

But what if Cass said "I love you" and Lucinda didn't reciprocate? That would probably result in some degree of discomfort for Lucinda, and a huge loss for Cass, especially if Lucinda was discomfited enough to decide to move out: they had both agreed that the arrangement was experimental.