36 Arguments for the Existence of God (26 page)

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: 1
¹
, 2
¹
, 3
¹
. In the second triangle, his red line had the first seven numbers raised to the power of 2—1
²
, 2
²
, 3
²
, 4
²
, 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: 1
³
, 2
³
, 4
³
, 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
n
th difference of
x
ⁿ
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 Rebbe–to–be, 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
n
th difference of
x
ⁿ
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.”

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.