2

I suppose it’s time to explain exactly what it is that happens to Urakawa-san. Of course, by that I mean the times where she shimmers unsteadily and is replaced by a version of her from another time.

It’s an automatic phenomena, and Urakawa-san can’t control it. We have no way of knowing when it might occur. It can happen at any moment of any day, all year round. She says that it happens about once every three days, and it’s always preceded by a headache.

We’ve taken to calling this condition “Substitution”.

For a short period, Urakawa-san will be substituted by a version of herself from another time. It can be either her past or future, and she never knows from when it might be. There was one time where we were meeting in the summer and her substitute was wearing a scarf.

It’s a complete substitution, so Urakawa-san never meets her other self. Evidently, there could only ever be one Urakawa-san in existence at a time. There was an interesting exception to that rule, though it didn’t come in the way I would have expected.

It was about three years ago, our first summer as junior high students. Urakawa-san shimmered and disappeared just as usual, but nobody showed up to take her place. For a time, I thought that Urakawa-san had disappeared for good. I was completely convinced. Then, a moment later, she came back exactly the same as before, as the present Urakawa-san.

“What the heck just happened?” I asked.

“The obvious happened, just as it should have,” she answered.

I couldn’t make any sense out of her response. Every time I brought it up after that, she got cagey, so I could only assume that she didn’t know or didn’t want me to know.

✽

I typically spent my days of summer vacation in the attic with her. That day was no different. I was lying on my stomach, using a cushion that I had brought to prop myself up as I worked on a puzzle.

It was a really weird jigsaw puzzle. Every single piece was white, with no distinctive illustration, and there were 1000 pieces. All put together, it would make about a 30×20-inch white rectangle. Urakawa-san had brought in the puzzle some three years ago. At first, I had no idea how anyone could even begin to solve such a puzzle, but we had already put together about 80 percent of it.

The two of us had agreed to take turns working on the puzzle, and now it was my turn. Urakawa-san was reading some kind of hardcover book while resting against the wall. The book had a script I couldn’t recognize written on the front cover.

I heard a, “Tsumiki,” and turning towards Urakawa-san, I saw that she had been substituted by a younger version of herself, looking to be around late elementary school age. There had been a recent pattern of her being substituted by younger selves. Her hair was wet, which caught my attention.

“How old am I in the present time?”

She would ask, I would answer. We’d had this exchange countless times. “You’re 15. You’re a first-year in high school, and it won’t be your birthday for a while.”

She nodded along to my explanation, then asked, “What on Earth are you doing there?”

“It’s a highly productive activity that uses just a few pieces of cardboard to grant incredible satisfaction. You’ll understand by the time you’re in middle school.”

“Do we have a good relationship?”

“Well, I think we do, at least. What do you think?”

“I think that we do.”

“Glad to hear that. I was worried that there was a secret part of you that didn’t like me. Have been for a long time.”

She nodded again, then faded away. Next, there was a first-year high school girl in front of me. “I also believe that we have a good relationship,” she said.

I bit back a bitter smile. “You sure have a good memory.” She could remember such a short conversation from so many years ago.

Every time Urakawa-san went through a substitution, she would pull out her memo pad and pen, scribbling down the details of the substitution. They all followed a general outline, typically ending up like, “Past, 11 years old, summer.”

I reached for a new puzzle piece, keeping her profile in the corner of my eye. I tried the piece in three places, but it didn’t fit in any of them.

Urakawa-san finished her writing, reaching back for her hardcover. I heard the sound of a page flipping, then she made a small noise. It was unfocused, somewhere between an, “Ah,” and an, “Oh.”

“Something wrong?” I asked, pulling my face up from the puzzle.

Urakawa-san was staring blankly at her hand. More specifically, her right middle finger. “I seem to have cut my finger on the page. It is of no consequence.”

I got up, heading her way. She was only three steps away, and I spent that distance reaching into my pocket for something I knew I had on me. “Maybe not, but it’ll hurt when you try turning more pages.” I pulled out a yellow Band-Aid. I had gotten it from the bookstore that morning. I doubted even the world’s best detective would be able to uncover why they were giving out free yellow Band-Aids with the purchase of a mystery paperback.

I took her hand, wrapping the bandage around her fingertip. I would never have had the chance to touch her without the excuse of her injury.

“Thank you very much,” she said, staring intently at her fingertip. She was so focused that, had she been looking only a few inches lower, I would have assumed her to be trying to palm read her future. “There’s a cartoon bear printed on this.”

I moved back to the puzzle, picking up a few pieces. “Oh, really? I wasn’t really looking. I got it for free, anyway.” I pressed down a piece, and it neatly slid into place. There was a certain satisfaction in fitting just the right piece in just the right place. Something about it helped me believe in the fundamental good of human nature. “It’s your turn now.”

Urakawa-san came towards me, tucking a bookmark neatly into her book. I laid down on my back, still using the cushion as a pillow, and picked up the paperback I had tossed to the side.

“I have something rather important to discuss,” Urakawa-san interjected.

“What’s that?” I asked, picking myself up.

“I don’t believe this puzzle will have all of its pieces.”

I looked over at the incomplete white rectangle. “What, did you count all the pieces?”

“No, but I had predicted as much seven years ago, while having this band-aid on my finger.” She waved her right hand at me.

“Well, that’s a problem.”

We had already spent too much time on the puzzle to watch it go to waste. Granted, it was just gonna be a white piece of cardboard, but I was still looking forward to it. There was even a small part of me that thought maybe some miracle would happen when it was put together, like a beautiful goddess would spring out of the board or something.

“And one more thing,” she said, counting the puzzle pieces. “I am likely going to disappear soon.”

I thought of a story Urakawa-san had once told me as we counted pieces. It was from her second year of elementary school. Back when she still lived in the very house we were in, but with her family. A time when she had a kind mother and quiet father, a time that almost seemed mythical and fairytale-like.

One winter evening, Urakawa-san got into a car with her parents to drive to a nearby restaurant. Her father was the driver, Urakawa-san was the passenger, and her mother sat in the backseat. They were just about to get off the highway while passing by narrow residential streets. They stopped at a red light, moving forward when it turned green. She heard a loud horn, and everything went black.

Urakawa-san woke up on a hospital bed. Her memory of everything that followed was reportedly fuzzy. Mainly, she remembered the same story that she was fed over and over again. Their car was T-boned by a large vehicle, and by the time the ambulance made it there, her parents weren’t breathing.

She remained in the hospital for a full month. By the time she was discharged, her parents had already been cremated. She couldn’t even attend their funeral due to the intensity of her head injury.

She went to live with her grandfather, who she described as particularly indifferent to his surroundings. Her grandmother had already passed, which left her living alone with her grandfather, a relative she wasn’t very close with. Her first night there, she sat alone in the corner of her room, her head swirling with thoughts she couldn’t control.

First was the absence of her parents. Then came death. That was followed by wondering if she could cry, before asking why she was unable to. Finally, she had the thought that wherever she was didn’t even feel like reality.

Is this like any other night I’ve ever known? Is this the reality that I’ve experienced so far? Am I even the person I thought I was? Everything felt wrong. White lost its brightness, and black lost its shades.

As far as I can imagine, Urakawa-san was simply overwhelmed. She couldn’t even cry because everything around her had become so incomprehensible. All she needed to become a normal girl again was for something to become recognizable and understandable. That just seemed like what she needed to me. But she had begun to dissociate from herself. She didn’t know if she was where she belonged, or if what she experienced was even real or meaningful.

I tried to imagine it. A lone Urakawa-san, a second-year elementary schooler, huddled in a corner on a winter night. She sat there, her face expressionless, staring blankly at the wall. Then, she put her hand to her forehead.

She said that at first, it felt more like a trembling sensation than pain. She imagined it was what drunkenness felt like, then it exacerbated into a splitting headache, eventually becoming so severe that she blacked out.

She was convinced that everything was over at first. She told me that she thought that was the end, and that it was only fitting. She wasn’t saddened by the idea.

But she wasn’t gone.

She realized that her surroundings had changed. She was back in the Western house where she grew up. Hot summer light shone in from the window, and cicadas chirped. The floor was covered in a thin layer of dust, and the house looked completely empty.

It was the perfect moment for her to feel sadness, loneliness, or nostalgia surrounding the place where she used to live. Perhaps she could have even cried. But she felt nothing. The house didn’t even feel like somewhere she was supposed to be. Then, before she knew it, she was back in the room of her grandfather’s house.

It was Urakawa-san’s first experience of substitution.

The present Urakawa-san remarked, “I knew it. We are missing a piece.”

No matter how many times we counted, we only ever came up with 999. By then, I had finally recalled my first experience of Urakawa-san’s substitution. It had indeed been seven years prior. She had appeared wearing a yellow Band-Aid on her right middle finger with a bear printed on it, saying a few sentences.

I’m worried about whether I’ll be able to find the piece that was lost.

She picked up a puzzle piece. It was 1/1000th of the full puzzle. Just like the piece we were missing. Her face remained ever expressionless as she said, “I’ve been theorizing about something for a while now.”

She poked at the puzzle piece. With her long, slender fingers. With her ovular nails. “You’ve probably noticed it too, right, Tsumiki? Whenever I shimmer away into substitution, my age ranges from elementary school to high school. The lower limit is eight years old, which was right around when I was substituted for the first time. I doubt that’s a coincidence.”

That fact had been on my mind, if only because the younger Urakawa-san always checked her age every single time. I watched her try to place a puzzle piece on the edge of the board. The shapes seemed to be similar, but it wouldn’t fit in place.

“The question is, what’s the upper limit?” she asked.

My response caught in my throat.

Urakawa-san put down the piece she had been working with and picked up another.

I carefully exhaled so that she wouldn’t hear and gave her my answer. “As far as I can tell, the upper limit is 15 years old.”

Or to put it another way, her current age.

“I will admit, I have been considering that for a long time. After all, why had I never been substituted with any older ages?”

There was a part of me that didn’t want to hear her say any more. But she continued.

“I suppose when the time comes, I will simply disappear.”

“Disappear?”

“I mean that the end will come for me.” She stared around, looking for another piece. “I am unsure what will happen when the end comes. Perhaps I will die in a traffic accident, or unknowingly catch a deadly disease. But given the strange reality of my power, it is possible that I will simply disappear.”

Urakawa-san was unstable, like a ghost. A simple disappearance seemed to fit the best for someone like her. But the possibility that she might disappear and never return became real and terrifying to me at that moment, like the lurking fear of walking alone at night.

I shook my head. “We don’t know that. Maybe your future self just hasn’t come around many times.”

“Do you really think that is the case?”

“Well, the possibility isn’t zero.”

“But it isn’t very far from zero.”

I could only nod helplessly in response. After all, with one change every three days, how many times would she have swapped in seven years? Somewhere around 850 times, at least. Her proficient notetaking was definitive proof against my claim.

“Do you remember seven years ago? I appeared in front of you wearing this Band-Aid. I specifically said, ‘This is the upper limit,’” she said, her tone completely offhanded.

I nodded. I had forgotten, but her bringing it up made the memory rise from within me.

She continued, “That must mean that my substitution had already ended at that point.”

The end of her substitution. “How could your future self have possibly known that?” I challenged.

“It would be easy. If I had never substituted with anyone older than 15, that is, older than I am now, that would be the limit. Keeping track of the numbers makes it self-evident.”

It really was that straightforward. Suppose that the nine-year-old Urakawa had substituted with her 14-year-old self. In that case, when she turned 14, she would have to substitute with her nine-year-old self in exchange. Her past and future selves had to substitute each other an equal number of times.

“According to my notes, there are only seven unfulfilled future substitutions remaining.”

Only seven more. After that, she could no longer substitute with her past self. There would be no more precedences of future substitutions. It would be the end of her substitution.

“But running out of substitutions doesn’t inherently mean you would have to disappear. What if you just returned back to being a normal girl?”

An Urakawa-san who didn’t suffer substitutions, who was no longer unstable.

“But I cannot possibly imagine any other outcomes. Besides, I have a sort of evidence.”

“Evidence?”

“It is quite likely that I have experienced the time following my disappearance before.”

I wished that I didn’t know what she was talking about. But I knew exactly what she was referencing. The time three years ago, when she disappeared but was not substituted.

“Perhaps I didn’t exist in the time I had been substituted to. That might have been why the substitution was incomplete. It might be that what I experienced was what came after my future disappearance.”

After that moment, she had told me, The obvious happened, just as it should have. Thinking back, that day had marked a turning point in Urakawa-san’s behavior. She had begun growing her hair out, then introduced me to a 1000-piece white puzzle.

I shook my head. “But I don’t want you to disappear.”

She offered me a rare, gentle smile. “I suppose it is possible that your words would keep me from wanting to leave. Perhaps I might stay here, my heart anchored by your wish.” But keeping her lovely smile that so easily moved my heart, she continued, “My only regret would be this puzzle. I’d like to complete it, if at all possible.”

I took a quick breath in and out, slowly pacing myself to find the right words. “I don’t know about you going away or anything, but I’d like to finish the puzzle, too.”

Urakawa-san’s smile vanished, but her typically blank expression seemed to be colored with relief. “Well, if we want to do that, then we must locate the piece that we lost.”

“Well, maybe if we contact the company that made the puzzle, they’ll send us a new piece.”

“Yes, but we would have to specify the column and row of the puzzle piece. We cannot know where the piece fits until we solve the puzzle. But we also cannot assume that I will remain here long enough for us to complete the puzzle and call in the new piece.” After a breath, she added, “I must say, I find that the beauty of puzzles is in how the cardboard can break apart and be put back together.”

I nodded. “Well then, let’s figure out that last piece.”

“Yes, as fast as we can. Although I am quite sure we will not discover it until at least after the day that I substitute with the me that had just met you.”

That was quite likely, as her substitute back then had spoken of finding the last puzzle piece. She had said I’m worried about whether I’ll be able to find the piece that was lost.

“At least we know you can’t disappear before then, either.”

The data confirmed that she had seven substitutions remaining. If she substituted once every three days, that would give us somewhere around 20 days.

But she shook her head. “Over the last two weeks, my rate of substitution has increased.” Her face was ever expressionless. “I have recently been substituting every day.”

I frowned. “Seriously?” I muttered instinctively. Of course, it wasn’t like she had a reason to lie about something like that.

“It may be summer vacation, but we are not together at all times. Many of my substitutions have taken place late at night or early in the morning. You could not have possibly noticed.”

But I should have noticed anyway. Especially when getting to know her better was one of my goals.

“Now then, we should begin searching for the puzzle piece,” she finished.

We started in the attic. We searched high and low, once looking as close as possible, and once looking from a distance. We even looked under the frame of our partially completed puzzle, taking care not to break it apart.

We moved out from the attic bit by bit. The room leading to the attic, the second floor hallway, the entire house…

But the puzzle piece was nowhere to be found.