When you go into surgery, they have you count backward from 100 as the anesthesia kicks in. 100, 99, 98… and you’ll usually be out by the time you get to 95.
At least, that’s what it’s like in the movies. I wouldn’t know. I’ve never had surgery.
Last March, I met with a neurosurgeon to discuss the possibility of surgery on my spinal cord. My recent MRIs had shown two thin syrinxes – cerebrospinal fluid-filled cysts in the center of the spinal cord – and if Google was correct, those syrinxes explained everything. Surgery to drain the fluid would be highly invasive and require months of recovery, but if it went well, I could be nearly symptom-free afterward. For the first time in years, ever since the pain crept in slowly at the age of thirteen, there was hope. Once this nightmare was over, maybe a year or two down the road, I’d just be a normal twenty-something, pursuing my dreams and feeling okay for the first time that I could remember.
But none of that happened. After I’d waited three hours for my appointment, trying to do homework in the waiting room as the weight of my future sat heavy on my chest, the neurosurgeon took one look at my MRI and told me the syrinxes were too small to be causing any symptoms at all. The appointment lasted five minutes.
Back to square 1.
When I’m awake at night, sometimes I try to count backward from 100. I want to fool my body into thinking it is being anesthetized. But I always get distracted.
When I was a child, I counted primes. When I got bored of that, I counted them in Roman numerals, then in Roman numerals in sign language. But now, I find myself passing numbers through the 3n+1 algorithm. If the number is odd, multiply by three and add one. This gives an even number. If the number is even, divide by two. The Collatz conjecture states that this algorithm will always bring a number back to one. Or, when continued past the point of one, a 4 --> 2 --> 1 loop. The path it takes to get there, however, may be quite a ride.
So instead of counting backward from 100 directly, I put each number from the Collatz algorithm: 100 --> 50 --> 25 --> 76 --> 34 --> 17 --> 52 --> 26 --> 13 --> 40 --> 20 --> 10 --> 5 --> 16 --> 8 --> 4 --> 2 --> 1.
It works.
So I keep going: 99 --> 298 --> 149 --> 448 --> 224 --> 112 --> 112 --> 56 --> 28 --> 14 --> 7 --> 22 --> 11 --> 34 --> 17 --> 52 --> 26 --> 13 --> 40 --> 20 --> 10 --> 5 --> 16 --> 8 --> 4 --> 2 --> 1.
Because 3n + 1 turns an odd number even, the next step is always a divide-by-two. What goes up must come down. I imagine that pain must be like this too -- temporary. Increase must always be followed by decrease. The decrease may never restore me to my pre-increase state. But it is, nonetheless, a decrease.
I continue. 98 --> 49 --> 148 --> 74 --> 37 --> 112 --> 56 --> 28 --> 14 --> 7 --> 22 --> 11 --> 34 --> 17 --> 52 --> 26 --> 13 --> 40 --> 20 --> 10 --> 5 --> 16 --> 8 --> 4 --> 2 --> 1.
And then I reach 97. That one’s a doozy.
For reference, the graphical representation of the path that 97 takes to get down to 1 looks something like this:
(change slide)
I am calculating it, but I suddenly realize it’s begun to loop. It’s not supposed to loop until it gets to 1. Something is wrong. I recalculate, but it’s following the same path. I worry that I won’t ever bring it back down again.
The Collatz conjecture has not been proven. Every case up to two-to-the-sixty-eighth-power works, according to software designed to brute-force countless calculations. Additionally, Riho Terras proved in 1976 that the Collatz conjecture holds for “almost all” numbers – a result that sounds disappointing to any non-mathematician but is, despite the apparent imprecision of this explanation – a significant result. But the question of whether it always will, for any number, has yet to be answered. I think about that now, and in my pain-addled brain I wonder if somehow 97 has been missed as a counterexample. I continue my calculations, my heart sinking as the numbers climb higher and higher, until finally, they start to fall.
Slowly. Oh so slowly.
And now, after 118 steps, we are back to 1.
I can handle just about anything, as long as I know it’s temporary.
That’s why I get such a thrill out of the 3n+1 algorithm. Even in cases like 97, when the numbers soar as high as 9232 before dropping, rising again, dropping again, I know it’ll come back to 1. Somehow. It might be hell to get there. But I know it will happen eventually.
The Collatz conjecture was proposed first by Lothar Collatz in 1937, two years after he received his doctorate. He was only 27, not that much older than I am now. I wonder if he, like I do, felt every so often that he was running out of time.
I keep calculating.
96 is easy. 96 --> 48 --> 24 --> 12 --> 6 --> 3 --> 10 --> 5 --> 16 --> 8 --> 4 --> 2 --> 1.
Encouraged, I move on to 95. 95 --> 286 --> 143 --> 430 --> 215 --> 646 --> 323 --> 970… this isn’t looking promising. But as I go, I see a familiar number – 364 – and I realize the rest of the sequence can be borrowed from my calculations on 97. There is familiarity. Though there are ninety-four more numbers in the sequence after 364, I already know what they are. There is safety in these numbers.
94: eight steps to get to 98, which I’ve already calculated. Twenty-five more steps to get to 1.
93: three steps to get to 70, which I saw in previous calculations. Fourteen more steps to get to 1. Easy.
I’ve already seen 92 show up while calculating 97. Seventeen steps to get to 1. I’ve seen 91 as well. Unfortunately. It makes up the bulk of 97’s sequence. But still, despite the fact that the numbers soar close to 10,000 before slowly, haltingly, returning to 1, I know the path it will take. It is familiar. Safe.
90: 90 --> 45 --> 136 --> 68 --> 34. Repeat. Thirteen more steps to 1.
89: 89 --> 268 --> 134 --> 67 --> 202 --> 101 --> 304 --> 152 --> 76 --> 38 --> 19 --> 58 --> 29 --> 88 --> 44 --> 22 --> 11 --> 34. Repeat. Thirteen more steps to 1.
88. Repeat. Seventeen steps to 1.
And so it continues. I piggyback off the work I’ve already done. So much of the work is not new. Yet I still write every number, relishing in its familiarity.
What goes up must come down.
It will always come back to 1.
I care less about my computations as I go. Rather than seeking 1, I seek the familiar number that I know will lead to 1, and I let the algorithm simply do the rest. Is this apathy? Is this giving up?
Mathematicians are discouraged from working on the Collatz conjecture at all.
Some believe it is an unsolvable problem. “We will never be certain,” some claim. “We can’t find a counterexample, but it may be unprovable.” Twentieth-century Hungarian mathematician Paul Erdos stated, referring to the Collatz conjecture, that “Mathematics may not be ready for such problems.” We can trust it – it works for any number that we could possibly care about, if indeed we care about the algorithm for individual cases at all. Yet we can’t trust it entirely, because it remains unproven.
There are two types of counterexamples that could prove the Collatz conjecture wrong. One would be a number which shoots off into infinity, never to return. The other would be a number that eventually forms a closed loop other than 4 --> 2 --> 1. Neither have ever been found. Every example tested has acted as expected.
60 takes three steps to become familiar. 59, two. 58 is familiar. So are 57, 56, 55. 54 takes two steps. So does 53. 52 is familiar.
Statistical analysis is useful for understanding how numbers going through this algorithm eventually return to 1. 3n+1 increases each odd number more than dividing by two does. But every odd number immediately becomes even after 3n+1, while not every even number becomes odd by dividing by 2. Some numbers smoothly divide by 2, uninterrupted by a single odd number, until they reach one. On average, each step of the algorithm multiplies the number by a factor of ¾, which is less than 1. Thus, on average, the number is always decreasing. This comprises the heuristic argument that the Collatz conjecture is indeed true.
But averages mean little on such a small scale. They also prove nothing, much to the disappointment of disillusioned mathematicians around the world, mathematicians who would love to solve this mystery, subdue the algorithm, tame the beast. But the Collatz conjecture submits to no one.
And neither does pain.
51 takes five steps to return to the familiar. 50 is familiar. 49 takes one step. 48, five. 47, two.
The 3n+1 algorithm is erratic. It is unpredictable, unreliable, and yet somehow always – by a non-mathematical definition of the word “always” – manages to descend to 1. I think this is why it so fascinates me. The unpredictability of each step is easily ignored when certainty of the final result can be obtained.
46 becomes familiar in one step. 45 in two. 44 is familiar. 43, one. 42, two. 41, four.
When I begin each day, I know that I will also end it. What I will suffer in between now and then remains to be seen. But I will reach the end, and I will crawl into bed and maybe sleep, probably sleep, eventually sleep. Every day, this happens. I trust this pattern. It has always worked before. I have tested it on 8353 days so far. Some days, it seemed less certain than others – certainly, the day my car was T-boned by a pickup truck tested my trust in this pattern far more than other days – but it holds.
Unlike the Collatz conjecture, however, this pattern is proven to break. One day I will begin a day that I will not end. Maybe the 8354th case will break the pattern. Maybe 8354 is my limit. Yet I continue to trust, instinctually, that it is not, that it will not, that I will end tomorrow and the next day and the day after that. I depend a lot more on this pattern that is proven to break than I do on a conjecture that may very well be always true.
Someday, it will end. But not today. Or so I trust.
There is nothing new under the sun. Or, at least, under 40.
Everything is immediately familiar. The numbers ascend and descend in their erratic patterns, but I know them, and I trust them. They all come down to 1. Every time. Down, down, down they go.
There is nothing new to be said about pain. It’s all been said before.
When I trace the numbers on paper, I am relieved to be nearing the end. But when I am counting backward to lull myself to sleep, hitting 5 makes me nervous. I am running out of numbers. I am running out of time to fall asleep. I am not anesthetized. I will have to start over, perhaps higher – 1000, maybe, to give myself enough material to ponder to finally drift off to sleep. But even in spite of this, I find the narrowing down of each number comforting. They approach 1, their paths growing simpler.
5 --> 16 --> 8 --> 4 --> 2 --> 1
4 --> 2 --> 1
3 --> 10 --> 5 --> 16 --> 8 --> 4 --> 2 --> 1
2 --> 1.
Yet even when they reach 1, the pattern need not end. It loops. And it can loop forever, if need be.
1
--> 4 --> 2 --> 1
--> 4 --> 2 --> 1
--> 4 --> 2 --> 1
Four. Two. One.
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