I stared out the window, streaking a line through the dust with my index finger. Outside birds chirped their secret language and rustled the boughs of trees, while passersby mouthed chatter on the sun-beaten pavement. A squirrel darted out from the flower bushes and kicked up clods of dirt onto the empty schoolyard. Another chased after it, as if hoping to catch the former's bushy tail, although I had no clue what it would do with it. I wondered if it would succeed as I traced their spiraling movements for lack of better diversion.
Well, not for lack of a better diversion. Class lunchtime hummed around me, as my classmates mumbled unintelligible thoughts into each other's ears and laughed at unintelligible notes of humor. I sat with the window view, as I hoped to do in my later executive life, and I sat in the front, as if to accentuate the theme.
Propinquity to the instructor always intimidated ordinary students, and they clustered in their social groups towards the back away from me. It suited my purposes well enough. On an ordinary day, I would have joined Sasahara and Koharu and Michiru in their tittering circle in the back. When I didn't, they understood the signal well enough. Truth be told, no ordinary day had passed since I had confessed my feelings to Takasaki. Since the government arranged the girl of my dreams as my future wife. Since my dream came true.
And turned into a nightmare.
If I hadn't confessed to her right before getting the government notice, would that have attenuated the awkwardness? If I had simply played it off as chance — not as chance, as the methodical, purposeful design of the Yukari algorithms?
I recalled the sight of her lustrous gemstone eyes tearing up that night, and I scolded myself. She looked so beautiful while crying, as terrible as it sounded. Not to discount the unreal pulchritude of her smile. Her famous smile, the Misaki Takasaki the world knew: the ever cheery popular girl who befriended everyone. The angel who could shine away a rainy day with just the right curve of her magazine-covergirl mouth, who had a sunshine aphorism or a motivational speaker's note of encouragement for every dour occasion, who could look into one's eyes for the first time with all the instant understanding and tugging string of connection of a years-long friend.
The real Misaki Takasaki wept in lonely tragedy, holding together the shards of her exposed, torn heart in desperation and whatever secrets threatened to spill out from within. Like a shattered diamond, any of her facets would have looked beautiful. And yet the authenticity of her true self, peaking out from beneath the cracks of the porcelain mask she had constructed for herself, lent a certain additional quality above the others.
"Takasaki-san, what makes you so sad?" I thought to myself, careful not to mutter it aloud.
I wished I could kneel beside her, hold her chin, and extract whatever words balanced forever unspoken on her lips by pressing my own. I thumbed over my dry lips as the latter twitched at the thought.
I rolled a rice ball across my lunch box with a chopstick, although its stickiness prevented it from getting far. Lunch had already half-expired and I still hadn't touched a single grain. A snack for afternoon English class, I supposed. Another bad habit developed in the aftermath of the government notice.
I wanted the old Takasaki back. The distant, optimistic angel, the unreachable if gorgeous scenery of my life. Even as I conversed with her, even as I went on dates with her, even as I held her hand in mine, she never seemed more far away than now. I never wanted to see Takasaki cry. I wanted to return that idyllic ambiguity, that dreamlike if unrealized possibility, without an explicit "no". To have confined my relationship with Takasaki to the perfection of theory rather than the hollow reality dragged and willed into existence by the Yukari system.
"Hey," a hesitant voice greeted, extending a hand in front of my face. I bolted up, chair screeching behind me, and shook it in return. He had an unkempt mop of brown hair, a few careless strands sticking out from the back, and his neck had a slight forward hunch. His face was a tad too long and I was not sure I had ever seen such wide-open, clueless eyes on a guy before.
Thoroughly average.
"Yukari Nejima," he introduced himself. The name didn't ring a bell. And what a strange moniker: named after the Yukari system? Weird parents.
"Firstname Read - " I started, but he cut me off.
"Oh, I already heard of you, plus Takasaki-san mentioned you were really good at math. I see your names at the top of all those exam and competition lists."
"Oh, yeah," I exclaimed as I recalled where I had heard his name before: in that exchange between Ririna and Takasaki at the chess tournament. "It sounds like you're acquainted with her, no? Are you in her class?".
"Yeah…" Nejima's eyes darted away briefly, but he covered it up with a grin and a nervous chuckle. I chalked the momentary lapse up to his apparent awkwardness and perhaps, as my modesty didn't want to admit, the intimidation of my reputation. "I'm in her class."
He handed me a sheet of paper with a bunch of equations on it.
"I'm…um, I'm stuck on number 3…do you think you could…lend me some help…please?" he stuttered out, shaking.
I glanced at the problem.
"True or False and Why?: For n2, the Gale-Shapley algorithm can take n^2 iterations or longer to stably assign between n men and n women."
I set his homework down and glared back up at him. "You said you share class with Takasaki-san. She's in calculus right now, why are you giving me a pre-calc problem?".
"Oh…" Nejima rubbed the back of his neck. "She takes math separately from the rest of the class because she's ahead of us. You do something similar, right?".
"Yeah, I take my maths with the local university on Fridays," I replied, before it hit me. Everyone kind of looked the same beneath me, at least in academics, and I had long derailed off the typical grade school curriculum track. So when Takasaki came to me for calculus help I had just assumed that freshmen took that level normally. I realized I never thought of Takasaki in that way before. Oh, sure, I always admired her and thought her wonderful and talented, but not like that. Why not? Because she was a girl? Because she was beautiful and nice and that was good enough for me not to bother to find out what else?
"Maybe the Yukari system did have some reason to match us up after all," I pondered. "Why aren't you curious enough to find out yourself?"
I looked down at my clenched fists, and wondered how I could have condescended her when I had put her on a pedestal.
"How much else do the computers think we share? What else are they?".
"She really is amazing, isn't she?" I blurted out half-awake, before wincing when I realized I had said that out loud, then hiding that reaction on further thought.
"Yeah…" Nejima trailed off, looking off into the suburbia beyond the windows. He looked as if about to say something more, but he bit his lip.
"She's so popular, yet sometimes she feels like such an enigma. You sound like you know more, you know, as her classmate," I commented, hoping to drip-feed further crumbs from this Nejima guy.
"Anyway, back to the problem," he drove through the best he could with his tentative voice, pointing a finger at it. "I don't understand. I think it's true because the algorithm is big O(n^2), but that doesn't feel long enough to be a proof. I can't even begin to think about doing it by hand."
Normally I wouldn't help, but his snub of my query flustered me so much I latched onto the opportunity to move the conversation.
"Oh, actually, first thing is big O(n^2) is definitely not the same thing as running for exactly n^2 steps. Second, it might help to just do it nonconstructively," I answered. "First, you know the algorithm has to end eventually and produce a pairing, right? Just citing the proof from class?".
Nejima nodded. "Uhhh…I think I remember that much. It wouldn't make sense if it never ended, right?".
"So if you remember the algorithm, each of the men - could be the women instead, but let's say men - proposes to a woman each iteration, and the algorithm continues so long as someone gets rejected. When everybody is set, we finish and pair them off."
I flipped over my English worksheet and scrawled a grid on the back.
"To force n^2 iterations, we need at least n^2 - 1 rejections, one per day. Now, the worst case scenario is where all n men have to ask all n women. What's the maximum number of rejections the algorithm can sustain?".
Nejima laughed, his loose red tie swaying under his unbuttoned collar. "I guess it's n^2? If every man proposes to every woman? So true?".
"Not exactly," I interrupted. "We know Gale-Shapley must finish by its definition - if a girl has nobody until the last day, or they're stuck with a guy they don't like, the algorithm will force them to accept him by the end."
"That's a bit harsh," Nejima commented.
"You say that while we live under the Yukari system, you softie," I poked back. "Anyway, back to the problem, so each of the n women can only reject up to n - 1 men, for n^2 - n rejections, which is greater than n^2 - 1 for n bigger than 2. Because we don't have enough rejections, we can't make the algorithm last until the (n^2)th day, so the answer is false."
Nejima nodded. "I see…that actually makes a lot more sense! It never even occurred to me to try to think of the problem that generally! I was stuck trying to make up some example group of boys and girls with preferences."
"No problem."
"Actually, do they use this 'Gale-Shapley' in the Yukari system? The ministry, I mean?" Nejima asked. "Just curious."
"Hm, probably. There's also the Kuhn-Munkres algorithm, but I find it unlikely they use that one."
"Why?".
I sketched out a group of four stick figures and a bunch of lines and numbers between them. "Gale-Shapley maximizes stability. It's supposed to prevent rogue couples with mutual incentives to cheat. Alternatively, Kuhn-Munkres maximizes overall utility. It just seems more likely the Yukari systems uses Gale-Shapley, given it's mandate."
I added, "One side effect you can prove, in a male-proposer Gale-Shapley, is that men would get their optimal partner out of all possible in stable pairings, and women get stuck with their least preferred while acceptable partner. And vice versa in the female-proposer version."
"Hmmm…" Nejima hummed, pressing a thumb to his chin. His eyes drooped a little, as did the ends of his mouth. "Yeah, that sounds right actually. From what I've seen, I actually…I mean, I feel like that's the truth, actually."
"Hey, sorry if this is kind of personal, but feels kind of related. Have you gotten your Yukari notice yet?" I questioned.
Nejima froze, then defrosted with his arms and smile at awkward angles. "Huh yeah…her name is Ririna Sanada. She goes to a different school."
"Oh yeah," I recalled from the abortive chess match. "She's actually kind of cute, but she's a bit awkward, but I guess that's sort of cute too as long as it's not abrasive. Lucky guy, you. Hope it's going fine."
"How about you?" Nejima shot back. "You're always the best…best in exams, best in competitions, more ahead of us in math and science than we've been in school. Except maybe in the looks department, that's Nisaka's spot - "
I glowered at him for a second.
" - but that doesn't matter for the Yukari system.…if such an average guy like me…I really wonder what sort of girl they would assign to someone like you."
"Oh yeah, she's - "
A lump caught in my throat, and I coughed to try to clear the imaginary obstruction, to no avail. My assigned wife on a piece of paper and in some government record system, sure. But did Misaki Takasaki bear any other relationship to me, as a girl, as a person? I hadn't even talked to her since that bad date when she repeatedly, emphatically declared that she could never love me.
I couldn't name her as my assigned wife.
She wasn't my anything.
I didn't have her, I didn't earn her, and I didn't deserve her.
"Oh, is it something personal?" Nejima backtracked. "Sorry for asking. I…I get how complicated it can feel. Trust me."
Plus, I had one nagging suspicion. Not for this average-looking, average intelligence, forgettable nice character. Even Nisaka, the handsome, well-reputed other boy Takasaki had mentioned off-hand at the chess tournament, made more sense. An archangel like Takasaki had the whole world on offer before her.
But my one dumb, random, out of the blue question wouldn't mean a thing to him if my hypothesis proved false. A harmless toss in the dark that I might as well try, I assured myself.
"Hey, Nejima, this is kind of random, so ignore it if it doesn't make any sense and just forget I ever asked," I prefaced, "but did you ever give an eraser to anyone back in elementary school?".
Nejima tensed up, his teeth clenching. He jerked up his wrist to check his imaginary watch, swiveled on a dime, and bolted for the door.
"Oh, lunch is almost over, I have to head back to my class now! I wouldn't want to make Ta - my class rep have to write me up!" he called out behind him.
It really, really, really could not be. Maybe I had misinterpreted the whole last few seconds, like the broken measuring instrument that had led to the accidental "faster than light" neutrinos a few years ago, because the other conclusion proved so implausible.
The most beautiful girl I had ever seen, out of all the possibilities, fell for this awkward oaf?
Author's Note: See disclaimer in chapter 1