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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 à 2 à 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 à 50 à 25 à 76 à 34 à 17 à 52 à 26 à 13 à 40 à 20 à 10 à 5 à 16 à 8 à 4 à 2 à 1.
It works.
So I keep going: 99 à 298 à 149 à 448 à 224 à 112 à 112 à 56 à 28 à 14 à 7 à 22 à 11 à 34 à 17 à 52 à 26 à 13 à 40 à 20 à 10 à 5 à 16 à 8 à 4 à 2 à 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 – that when I curl up and whimper into my pillow, the world narrowing to nothing outside my body in the fetal position and the feeling that I am burning, it must be 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 à 49 à 148 à 74 à 37 à 112 à 56 à 28 à 14 à 7 à 22 à 11 à 34 à 17 à 52 à 26 à 13 à 40 à 20 à 10 à 5 à 16 à 8 à 4 à 2 à 1.
And then I reach 97. That one’s a doozy.
For reference, this is the graphical representation of the path that 97 takes to get down to 1:
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 268 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.
There are days that I wonder if I’ll make it. I wonder if this wave of pain will pass, if I’ll return to a new baseline that I can learn to live with. I wonder what will happen to me if it doesn’t. But it always does. I often wonder if I’m really going back to normal, or if every time the pain rises it settles at a level a little higher than it was before. I wonder if there is a limit to the pain that I can bear. I wonder if pain is cumulative. Is living like this for years harder than if I’d woken up this morning feeling like this when I’d been fine before? Or is it easier?
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 à 48 à 24 à 12 à 6 à 3 à 10 à 5 à 16 à 8 à 4 à 2 à 1.
Encouraged, I move on to 95. 95 à 286 à 143 à 430 à 215 à 646 à 323 à 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 à 45 à 136 à 68 à 34. Repeat. Thirteen more steps to 1.
89: 89 à 268 à 134 à 67 à 202 à 101 à 304 à 152 à 76 à 38 à 19 à 58 à 29 à 88 à 44 à 22 à 11 à 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.
The pain scale has always confused me.
Where the capacity for empathy is lacking, we must still find ways to understand, to quantify, another’s pain. We know what the numbers mean. One is fine. Five is unpleasant. Seven sucks. Ten is agony. If we ask someone to rate their pain, we can escape the need for empathy by quantifying their pain objectively. When empathy fails, how do we decide how much sympathy to afford another’s pain? The numbers reveal the answer. Three does not require sympathy. Nine does.
The problem is, the pain scale does not work for people like me.
Pain is simple. Pain is easy to quantify when your baseline is zero. But it gets a lot muddier when pain is normal.
87 is all new. 87 à 262 à 131 à 394 à 197 à 592 à 296 à 148 à 74 à 37 à 112 à 56 à 28 à 14 à 7 à 22 à 11 à 34 à 17 à 52 à 26 à 13 à 40 à 20 à 10 à 5 à 16 à 8 à 4 à 2 à 1. But 86 takes five steps to get to 98, which I know. Twenty-five steps to get to 1. And 85 is easy. Nine steps to 1. 84. Nine steps to 1.
83 becomes 94 in five steps. One hundred and five more grueling steps to 1. 82 becomes 94 in five steps. One hundred and five more steps to 1. 81 becomes 61 in three steps. Nineteen more steps to 1.
1-10 doesn’t provide enough room for gradients. For this reason, when I try in vain to rate my own pain, I prefer to think of the scale as going from 1-100. Only then can I properly appreciate the distinction between a 78 and a 73, a level of precision not offered by the traditional scale. 7.8 and 7.3 are invalid options. “It’s either a 7 or an 8 – choose one.” And unless you are vomiting, don’t say 8, or you will be disbelieved.
Sometimes, I wonder if pain has limits.
The pain scale requires an upper bound. Ten – the worst pain imaginable. A red-hot iron rod through my eyeball, barbed wire through my teeth, perhaps. This is the kind of pain that I fear, the kind of pain that indicates irreparable damage to body tissues, the kind of pain that leaves psychological scars along with physical ones. But the worst pain that I ever experienced was the result of lying still too long in an MRI scanner. It was meaningless pain, nothing to fear. Innocuous. But nonetheless, it broke me, leaving me sobbing uncontrollably in the machine and leaving motion artifact from my trembling on the resulting images.
I say this pain was meaningless, but it means everything. No wound, no external stimuli, hurts as much as just lying still. Pain exists regardless of external factors. Pain exists independent of fear. Pain can break me even when there is nothing causing it. If you do not regularly experience pain, any pain is a sign that something is wrong. Something requires attention. Pain is meant to mean that something is wrong. We are meant to react to it – to pull our hand back from a flame, to lift our foot off a thumbtack, to freeze and immobilize a broken limb. Pain is our body’s natural defense. We fear pain, which is the point, because it makes us avoid pain. But what happens when pain is normal? Can someone who is used to being in pain still fear it viscerally, as would one who does not?
The MRI showed next to nothing, by the way. After two and a half hours of counting down the minutes, waiting, hoping, begging it to end. The neurosurgeon said the syrinxes were innocuous. So there remains no answer.
My coworker taught me how to use the pain scale correctly, using his personal experience. “When I broke my back, that was a ten,” he told me, and went on to describe what it felt like. I don’t remember the words he used because all I could think was how familiar his description of this pain felt to me. I’d been there, or at least close. What I do remember, however, was this line: “Pancreatitis was an eleven, though. Absolutely off the charts.” The pain scale is limited to ten, yet he added an extra number to describe true agony.
80 is familiar. Nine steps to 1. 79 takes fourteen steps to 38, which is familiar. Twenty-one steps to 1. 78 to 202 in nine steps, twenty-six more to 1.
I rate my pain at a 77 today. Twenty-two steps to 1.
Every time I think I am used to the level of pain that I live with, it gets a little worse, and I have to readjust. I spend a few days, maybe a week, in agony, before I remember how pain works and I remember how to keep it under wraps. It increases by small increments every month, maybe two if I’m lucky. I tell myself it’ll settle back to normal soon – what goes up must come down – but it seems that instead, my capacity to deal with pain rises to the occasion. I grow stronger. I grow weaker too – my knee buckles more, my vision clouds – but I grow stronger of heart, of mind. Could I still be fighting for some semblance of a life if I hadn’t learned to minimize my reactions to the constant burning in my head and neck and shoulders?
My body still reacts, but it reacts all wrong.
I contemplate my limits, and I deny them. My capacity for pain is limitless.
76 goes to a familiar 38 immediately. Twenty-one more steps to 1. 75: fourteen steps to 1. 73 is hell. It takes twenty-three steps to get to 91, which is eighty-two steps away from 1. 72 is quick, comparatively – twenty-two steps – but 71 takes ninety-two. It’s a system in which 100 is closer to 1 than 71 is. 100 ends quickly, at the very least.
70 itself is familiar. Fourteen steps to 1. Easy. Painless, almost.
When I first began working with a physical therapist, she asked me every session what I would rate my pain. I usually said six. It felt melodramatic to rate it any higher. I was used to it. I could handle it. And besides, improvement was almost certainly out of the question. I just had to learn not to crumple when it hit a little harder than I was prepared for.
Instead, I indulged my melodramatic tendencies in the privacy of my journal, where I scratched out lines of “poetry” (a term I used loosely to refer to highly descriptive whining). “My jaw feels like rusty metal sounds/ When it collides with the car that/ I no longer drive.” “My hands tremble like sea foam/ Collecting on the shoreline.” I described my pain as feeling like burning rubber smells. Clumsy attempts to express what this pain is like, to describe it accurately, so that one who does not feel like this can feel it. But I only manage to obscure it even further.
Still contemplating the pain scale, as I often do, I wrote this poem to describe the difference between a seven and an eight. I ended with these lines:
Sometimes eight feels like dying
But still I answer “seven”
Until eight feels like seven
And seven feels safe
Again.
But really, at the end of the day, the difference between seven and eight means little. It’s all just numbers. Meaningless numbers with little gradient. Arbitrary, really – word games I play with medical professionals who truly believe that my answer matters, that my answer can determine what level of care I am owed.
69 takes three steps to become familiar. 68, only one. 67, only one. 66, five. 65 is itself familiar. 64 is an interesting case – a power of 2, it divides by 2 until it reaches 1, never requiring the 3n+1 step. 63, two. Nevermind that the erratic descent to 1 after that point takes up a full page of computations. 62, two. 61, two.
I was thirteen when I first noticed the pain. Something felt wrong. Life felt harder for me than it was for my sisters, who could effortlessly run around and play for hours without tiring. It seems unfair in retrospect, but at the time, I simply accepted it. This is normal, I thought. I avoided fear. I ignored pain. I didn’t want to face what my life would be like someday when it got worse. And when it did, when I had to face it, I closed my eyes like a child playing hide-and-seek. If I can’t see you, you can’t see me, I told the pain, and it smirked.
But I could trick myself sometimes. I’d run along the Lake Michigan shoreline, the water so cold my feet went numb, but it distracted me well enough. I’d play a game of ultimate frisbee and take countless dives into the ground, risking bruises I didn’t fear to score a point. These are the things I miss now. I do not run anymore. I barely dare to stretch my arms above my head these days. It is all too risky.
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 à 2 à 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.
I return once more to the question of limits. Like 97, 47 brushes 9232 before it returns to more manageable numbers. Is 9232 the limit of my pain? Or might I reach beyond and find a way to survive it still, even if it climbs to 9233?
Does 9232 mean anything at all? Does any number?
When will I break?
To understand my pain, sometimes I must observe it through the eyes of another.
I choose my friend Collin. He has, for one reason or another, been privy to how it breaks me. He has seen me break. I wonder what it looks like, if I look as anguished as I feel, or perhaps more so. Around him, I have allowed myself an ounce more melodrama than feels reasonable, or perhaps a pound. I put on a show, occasionally going so far as to writhe on the floor of his apartment, to hiss when he touches me while still grasping his hand and begging him not to let go. I don’t want to be alone with this pain. I need his touch, to remind me that I exist, that there is more to the world than the quilt of porcupine quills that someone has wrapped tightly around my shoulders and pressed into my face, expecting it to suffocate me.
I choose my ex-boyfriend James. “You know, with how you talk about it, I expected you to be basically writhing on the floor 24/7,” he told me once. Joke’s on him – I have, many a time. But not enough. In his eyes, my face is glacial. Steady. I smile cautiously, gently, showing nothing of what I’d told him I suffer from. “I can handle this,” he said another time, referring to my pain’s potential effect on the future of our relationship. He could handle it, perhaps. But I couldn’t.
I can’t handle this, still.
46 becomes familiar in one step. 45 in two. 44 is familiar. 43, one. 42, two. 41, four.
I choose my mother. She knows. She’s been there. My pain was never as bad as hers, until it was, until it was worse, until it broke me. Maybe she’s just stronger than I am. But when she saw me twitch and shudder and then freeze in place to stymie further twitching and shuddering, she thought I was going to die. At times, I thought so too. I know better now.
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 numbers failed, I reached for metaphors. Like burning rubber smells…but those fail too. My senses are all connected – the squeal of bus breaks feels like a brick to my head and blurs my vision. My allodynic skin feels the way that fingernails on a chalkboard sounds. My throat constricting feels the way lurid graffiti over garbage cans looks. The burning and trembling of my shoulders feels like inhaling turpentine. None of it makes sense. None of it makes sense at all.
The first draft of this essay read like a near-copy of another essay I read once. I had no idea until I read them together. I have nothing new to say, not really. It’s all been said before. Words are inadequate. The closest most of my friends will get to understanding my pain is when they offer their hand for me to hold, and I dig into their palms with nails with all the strength I have left – not much strength, not really, but just enough. I wear the pain in my eyes now and it just looks like me. It’s just how my eyes look. Wide sometimes, droopy sometimes. I hurt. It’s who I am now.
5 à 16 à 8 à 4 à 2 à 1
4 à 2 à 1
3 à 10 à 5 à 16 à 8 à 4 à 2 à 1
2 à 1.
Yet even when they reach 1, the pattern need not end. It loops. And it can loop forever, if need be.
1
è 4 à 2 à 1
è 4 à 2 à 1
è 4 à 2 à 1
Four. Two. One.
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