36 Arguments for the Existence of God - [68]
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: 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 nth 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.
