(Not) Random Ball-Drawing

I

Founders' Day, a warm and pleasant afternoon.

I ambled toward the gymnasium, humming a tune and swinging the sleeves of my school-uniform jacket. Every so often some unfamiliar face in a gaudy, multicolored class shirt would dash past laughing loudly, leaving me looking rather like a single dot of white amid a riot of flowers. This is probably a scene you can only catch once a year — the school rules are strict, and once you leave the dormitory the outermost layer must be the uniform. At most you may pin on the school crest; all the other flashy badges are forbidden, and dyeing your hair is even more strictly banned. Only on Founders' Day is everyone granted the rare exception of wearing their class shirts on the outside.

"Their class's shirt looks pretty good, but ours isn't bad either." A-Yu, walking with me in his sky-blue class shirt, bounced along, now and then puffing out his chest as if deliberately showing off the fancy figure 12 emblazoned across the front. "I really wish we could dress like this all the time. Or at least make the uniform a bit more distinctive. All white is just too dull." He thought for a moment and then griped some more: "And that unchanging school badge — the design's so old-fashioned, too. Especially that number on it; it's way too big, an eyesore, like some bottle of Guojiao 1573 liquor."

"That's exactly how you can tell we're a hundred-year-old school."

Considering that A-Yu belongs to the reformist faction with no shortage of complaints about the school rules, grumbling like this was hardly surprising. He was always going on about how this uniform, apart from being excessively formal — from a distance the crest looks like something a university would have — is otherwise just clean and blank, too plain and stuffy, and easily stained besides. Far better, he said, to open things up and allow free styling; failing that, at least copy our brother school next door and come up with a halfway decent design. Now that he could finally wear the class shirt that everyone had discussed and designed together, his mood was perhaps, in a sense, the same as that of getting to play on your phone after going home following a month of consecutive overnight stays at the dorm.

I, for one, didn't feel much about it; perhaps I'm simply the sort of person with no real individuality. To me, clothes are just a covering for the sake of decency — presentable enough is fine, and agonizing over them any further is pointless. Hairstyles even more so. The school's unifying everyone's dress to save thinking time is a truly excellent thing. And looking at it from another angle, when graduation comes, doesn't it leave more space for friends to sign their names?

"See, that's exactly your typical otaku mindset. It's precisely that attitude of yours that means no girl will ever fancy you."

"Don't take that preachy tone with me, it's so old-fartish!"

After a string of harmless banter, our destination — the gymnasium — was finally in sight. As for why the two of us, on this bustling Founders' Day, had strayed from the main battlefield of the class stalls and exhibitions to discuss such empty, festival-clashing topics while running off to this out-of-the-way corner, well, that goes back to something the class committee said last week.

"Next week there's an activity called the 'Fun Relay Race,' two people per class. It's purely for fun, not much technical skill involved. Want to take part?"

Perhaps because the centennial Founders' Day was rather special, this year a whole batch of odd new activities had cropped up out of nowhere. Given that the other classmates were already wearing several hats and short-staffed, the two of us — idlers with nothing to do all day — were quite naturally dispatched to it. We were happy enough about it, too; after all, this activity really was just like play, with none of the dreariness of being banished to some distant land, more like skipping class to go to an amusement park.

When a person is in a good mood, their luck usually isn't bad either — that's always been my belief, and today proved no exception. In the few rounds just gone, the two of us had, remarkably, cut and slashed our way through every match in first place all the way to the final, and were about to come over to this table-tennis room to meet the last showdown against two classmates from Class 10 and Class 5. And the final event would be…

Once again I unfolded that crumpled little slip of paper.

II

"Drawing balls at random? It's not playing cards after all? This feels even less classy than the earlier ones."

Without realizing it, I had read the rules aloud under my breath.

1. On the floor are two identical transparent boxes, one of them empty, the other holding three balls of different colors. Each ball represents a fixed point value, but only the referee knows it. In each round of drawing balls during the match — including any tiebreaker round that might occur — the order of drawing strictly follows the ranking from the semifinal.
2. The match is divided into a first half and a second half; each class sends one classmate to compete in each half. In the first half, the referee will first announce the number of rounds to be drawn. Then, in each round, the Player A of each class steps forward in turn to draw one ball and place it into the empty box, while the referee records the score; after one person finishes drawing, the next person waits 10 seconds before drawing. Once all three have drawn, the round ends (at this point all three balls have been transferred into the empty box, and the box that originally held the balls now becomes the empty box instead). Ten seconds after the third person finishes drawing, the next round begins, the first player continues to draw, and so it goes back and forth for a number of rounds, after which the first half ends.
3. The Players B of each class wait outside the venue, and may only enter the field to compete once the second half begins. The rules of the second half are exactly the same as those of the first half (including the drawing order and the number of rounds). After it ends, the referee tallies the total scores of the first and second halves to determine the final ranking. If a tie occurs, a tiebreaker is held; the tiebreaker consists of only one round, after which the final placings for champion, runner-up, and third place are decided.

Note: Before the second half begins, Player B may not enter the field to communicate with Player A. The total score may only be announced after the match ends; before then, the competing players also may not sneak a look at the scores.

"Sounds awfully complicated for what it is." A-Yu snatched the slip from my hand. "But when you boil it all down, it's just a probability problem — we don't know the point values of the various balls, so there's no way to know which ball is better to draw. Round and round it goes, but the probability of winning stays at one in three, independent of the drawing order. The few supplementary rules tacked on at the end are also there to ensure the result is completely random. After all, if you could see both the ball you drew and the balls others drew, and then see each of the corresponding scores, you could set up a system of equations and solve for the value of each ball. The two players not being allowed to communicate is the same logic — it's to minimize as much as possible the effect of any known information on the game, things like the number of rounds.

"In other words, this time it really is pure luck. The card game earlier had no technical content either, but at least it had some. Now it's truly down to fate." Babbling on, I muttered, "Speaking of which, for a game this simple at heart, why dress it up so fancily? The design's laid on awfully thick."

"Maybe it's to highlight how important the final is, without wanting to increase the game's complexity, so they deliberately made it all solemn and ceremonious. But if you ask me, this whole competition is plain stupid — though at our age we've done no shortage of stupid things ourselves."

As A-Yu talked on, he started sounding all grown-up again, so I hurriedly changed the subject.

"We ranked first in the earlier semifinal, so we go first, right?"

"Yeah, but like we just discussed, the first-mover advantage is useless in this game; it's still down to luck. The only upside is probably not being placed last, so you don't have to wait around for a good while and then symbolically pick from the balls others have left behind."

"Not necessarily. Drawing first won't change the probability of winning the championship, but it can guarantee that we don't lose too badly."

"It's barely even started, why are you already locking in a worst case? Can't you think about something positive, you guy?" A-Yu made as if to hit me.

"Oh come on, this is such an obscure competition that hardly anyone even comes to watch. It's not a bad thing to finally make the final and have a go. Listen, my suggestion is that in each round we draw a different color as much as possible. After all, if you keep drawing a single color, you might just keep getting the lowest score. Hmm, or how about this: you just draw in a fixed order — color A, color B, color C, then color A again… and so on. As long as, over n rounds, the number of times any one color is drawn is no greater than ⌈n/3⌉ (rounded up). Drawing this evenly should net us about a third of the points, so we won't be routed; we might even sneak into second, or even steal first."

"Same logic as your soccer betting over summer break, right? In your heart you were hoping Ronaldo would crush mighty France within ninety minutes and lift that damned Henri Delaunay Cup right away, yet you also put twenty yuan on France winning in regulation, while putting on a whole mysterious act for me about how you predicted it'd be a draw and go to extra time, guaranteed. Basically you cycle through all three possibilities once, so one of them is bound to hit, and either way you won't lose too badly."

"Correct. Soccer is betting, drawing balls is betting too…" I paused, then added, "And besides, doing it this way might also pile psychological pressure on the opponents. Think about it — what would the second person to draw think when they see it? With enough rounds they'd surely spot the pattern too, and might just choose the same safe strategy. That way we drag the opponents down to our level."

"What kind of crooked reasoning is that? You're thinking far too idealistically! Do you really take yourself for some battle-of-wits genius? You think you've got a protagonist's halo and everyone will just play along with you? In your dreams, peach."

Grumbling aside, A-Yu was won over by me and decided to adopt this intuitively can't-lose-too-badly plan of attack. Well, our luck earlier had been very good, but how much worse than us could opponents who'd made it this far really be? Betting purely on luck was still too dicey. Between a one-in-three chance at the championship and a high probability of not finishing last, the latter, as it turned out, was the more tempting after all.

At the time I had no way of knowing that it was precisely this decision that would turn the tables later on.

III

As Player A, A-Yu was soon called into the table-tennis room to compete. The moment he and the two from Class 5 and Class 10 went in, the door was shut tight and the curtains drawn fast, leaving not the slightest chance to peek.

With nothing to do, I leaned against the corridor wall and rested with my eyes closed, waiting for the whistle that would start the second half. I don't know how long passed before the side door of the adjacent equipment room suddenly creaked open. A girl wearing a referee's pass walked in waving a blank sheet of paper. She went straight past the equipment area — with its piles of various balls, plus a whiteboard, a scoreboard, and other auxiliary tools — came to a small wooden table by the window, carefully wiped the empty, dust-covered tabletop clean with a tissue, sat down with her back to the corridor, took out a pen she carried with her, and began scribbling something.

Was the first half over, and she recording the scores? The rules said no peeking, but I'm no upstanding type to begin with, so naturally I got up to something underhanded. I crept gingerly to the window and pressed up against it, peering inside, trying not to make too much noise, and shamelessly began to spy. But the paper was half blocked by the referee's slightly tilted upper body, and on top of that the writing was extremely small while my eyesight is only ordinary, so I had to adjust my posture several times in a row before I could barely make out that she was writing a few lines of crooked English.

What's this — turns out she was drafting an English composition. I was sorely disappointed, but still hoped she'd write something more. Just then, however, another girl, her hair tied up with both hands, gently pushed the side door open. My eyes instinctively darted away and down; I didn't dare meet her face, yet I had no intention of leaving. It wasn't until she had walked a fair way toward the wooden table that I came to my senses, hastily crouching down to pretend nonchalantly to tie my shoelaces, then bending over and shuffling in little steps to the other end of the corridor. How thoroughly embarrassing — but I shouldn't have been seen? Phew, close call.

After that whole fuss, I didn't dare press up against the window to spy anymore, yet I couldn't quell my curiosity, so I stood on tiptoe across the corridor and craned my neck to look. But because of the angle I couldn't see everything no matter what, and the voices, too, I could barely make out the gist of, not very clearly. It was at this point, though, that I noticed the girl was from our own class. Her name, however — oh no, I couldn't remember it. I thought again of A-Yu's gibe, "It's precisely that attitude of yours that means no girl will ever fancy you," and couldn't help lowering my head and sighing.

The newly arrived girl had brought nothing; she simply dragged over a stool and sat down directly facing the referee, lazily slumping onto it.

"Wow, do you really have to be this diligent? Even at a time like this you're squeezing in every spare second, not even properly listening to me. You're not telling me you write while you're keeping score too?"

"This is called making good use of fragmented time. I don't write while scoring, though — after all, I have to flip the scoreboard at every equal interval of time. Flipping the cards takes no brainpower, but if I really did write a bit, stop a bit, it'd easily disrupt the rhythm and break my train of thought. But you didn't come find me just to ask about this, did you?"

"Aiyaya, seen right through, am I. Then I'll just ask straight out: how many points does our class have right now? I'm just a little curious, that's all. I promise, promise I won't tip off the two from our class who are competing. A little gossip is no problem, right?"

"But you were just in there watching too, weren't you? Who knows your memory better than I do? You can still recall perfectly clearly who drew which ball in which round, can't you?"

"That's true, hehe, thanks for the compliment. But don't worry, how could I possibly cheat? Who knows my character better than you! Besides, you said this morning that your school crest had been lent out to borrow other equipment, but the balls and scoreboard used in this final were all borrowed using my school crest. Just consider it returning a favor…" The girl sat up straight and pleaded.

"All right, all right. Borrowing a few balls and you want a great big merit award? Looks like if I don't tell you, you'll keep making a fuss, so I'll tell you. There, it's this number." The referee lazily slumped onto the table, slowly extending one finger diagonally upward until it stopped in front of the girl's chest, and gave it the faintest of wiggles, twice. The girl across the table lowered her head, looked, and tentatively asked, "Ten…?"

The referee smiled and nodded. Only then did I realize that I had just been distracted and hadn't clearly heard the number the girl reported — I only caught the "ten" at the front, so I knew at least that it was "ten-something," but I couldn't be sure exactly what. Still, there was no way I'd seen it wrong: the referee had clearly extended just one index finger and wiggled it, with no other movement, and the girl opposite immediately guessed it was "ten-something" — so could it possibly mean eleven? Sigh, I'd eavesdropped just fine all the way up to here, so why did I have to drop the ball at this point? It's just like how in an English listening exam the most crucial part is always the bit you can't hear clearly, while you can pick up some information at the trivial spots. But I had no time to dwell on it — just as, even after missing something in a listening exam, you still have to grit your teeth and carry on.

"Eh, that little?"

"You're talking nonsense when you don't even know the point values of the balls. Actually it can't really be called especially little."

"Well I never. Eh, and by the way, how many points do the other two classes have? I tried hard to think but I still couldn't quite work it out…"

"What a coincidence — Class 5 has the same as you. Class 10, on the other hand, is way out in front, equal to the two of you added together."

So Class 10 had a full 22 points? No, no — I still couldn't be certain that gesture meant 11. But leaving aside exactly what "ten-something" it was, being this far behind was still, all in all, a total failure of the strategy. But that can't be right — how on earth did A-Yu manage it? Did he really listen to me? Following my plan, how could he possibly fall behind this badly? A-Yu isn't the type to change his mind at the last moment either. Was it because one of the balls had too disproportionately large a value…?

A half-formed idea suddenly took shape in my mind.

IV

The bell for the second half had already rung. The referee gathered her things and was about to leave, and I quickly walked into the table-tennis room, coming face to face with A-Yu, whose eyes were wandering.

"Did you act according to plan?" I mouthed the words at him, but he only nodded, then shook his head.

What's that supposed to mean — don't be a riddler! But then, the rules did say no communicating, so I had no choice but to guess. With no time left to keep up the encrypted exchange with A-Yu, and the referee urging me on, I had to go in cold. I took out a small red ball and placed it into the empty box, then, after drawing, tentatively observed the referee a few times. Behind the wooden table where the referee sat hung a clean whiteboard with a black marker and an eraser on it; on the wooden table sat the scoreboard (as shown below), but it was completely blocked by a low partition, her hand movements entirely hidden, so there was no way to roughly infer the size of the values from how fast or slow she flipped the cards.

I circled around to the back of the line and continued to watch closely the ball-drawing movements of the other two and the referee's reactions. The classmate from Class 5 drew a green ball, and Class 10 a white ball. Throughout, the referee kept her head raised high, watching us expressionlessly, and only when someone had drawn a ball would she quickly lower her head to glance, then raise it again. The card-flipping must all have been done blind, precisely so as to expose as little information as possible… eh, expose, information? A-Yu just now nodded and then shook his head…

I finally remembered: A-Yu seems to have red-green color blindness, which is why, when he saw classmates from other classes passing by in their gaudy, multicolored class shirts, he'd think they were all from one class. Back then he didn't seem to have recognized them.

If that's the case, then it's very possible he also can't tell apart the red ball and the green ball in the box. And the meaning of his nodding and then shaking his head was perhaps this — he tried his best to do as I said, but as to whether he had achieved the intended goal, he himself had no way to judge. After all, he drew in what he believed was the order red, white, green, but what he actually drew might have been red-white-red or green-white-green. If that's really the case… In an instant, those tiny, fragmentary clues arranged and combined themselves just so, constructing a road to certain victory. I held my breath and rapidly ran the calculations in my head. There shouldn't be any mistake.

When it was my turn to draw again, I no longer hesitated; after a deep breath I strode forward with my head held high and took out the very ball I wanted. After a number of rounds, the second half ended, and just as I had expected, I completed the comeback, narrowly overtaking Class 10 by the slimmest of margins.