One day, after randomly scanning down several rows of bookshelves, Emmet made a fascinating discovery! Within one of the massive shelves of formations related curriculum content, he spotted a familiar book - one with no words on it, only shapes and diagrams. Taking out his precious picture puzzle book and comparing the two, he found that they were virtually identical in shape and design - it was the same book!

After asking the receptionist about it, Emmet learned again that it was called Gemini’s All-Encompassing Formations Manual, a relatively famous book created by the one and only Gemini himself. The story was, Gemini himself came from a disadvantaged family that couldn’t afford to enroll their boy into a formations school. Thus, he had to teach himself the concepts, and had a difficult time reading through the only available convoluted books about them, full of terms only educated folks had been exposed to. Thankfully, he was able to be sponsored by a wealthy expert who had happened to discover him and valued his talents, and was able to deliver him to the then small, but now enormous Gemini Academy. After rising to prominence, he recalled his youth and wanted every gifted child, even ones who were never granted the luxury of literacy, to be able to study formations.

Thus, he spent two years tirelessly creating his All-Encompassing Formations Manual. In only all theoretically intuitive puzzles, diagrams, pictures, and simple labels, it contained virtually every single significant concept within the study of formations known within it, starting from the most basic and ending in over 30 pages of unsolved problems. In theory, were a child gifted enough, they could rely on pure intuition to solve a large portion the puzzles, without ever having to get bogged down with the mire of complicated terms and structures.

After the book was published, however, it received great ridicule. To many, it was Gemini’s lowest point. Why spend two years of your precious career writing this useless picture book? Did he really think that some illiterate commoner could suddenly start learning the complexity of formations if the barrier of literacy was removed? They commended him for the ability to condense all the concepts into puzzles and diagrams, but thought it was mindless to think that any commoner would actually be able to get anything out of it by themselves. Formations were absurdly complicated and sophisticated. You needed the context around it, and specialized tutors to build up understanding from a young age. There was a reason why nearly every formations expert was a noble!

Sure enough, after proliferating out throughout the world, nothing ever really came of it. As far as to the formations world’s knowledge, no commoner child had ever managed to use the book to thoroughly learn formations on their own. The receptionist himself had laughed, saying that it was a joke that any commoner would be able to create any understanding from it.

Quickly nodding, Emmet turned away and returned the book to its shelf. He quietly thanked Gemini under his breath. This manual had opened up so many doors for him! Even now, he felt it was much better than any textbook he had ever read. Gemini was truly a genius - every puzzle in the book was laid out in a perfect intuitive fashion, allowing someone to understand what the structure of the problem was with limited words.

He went back to his seat and dove back in into a textbook he had been reading.

Like this, two more months passed. With each week, the end of year assessment approached closer and closer. It hung over everyone’s heads like a weight.

However, Emmet didn’t have to worry about that. Since he was a squire, he didn’t need to take the end of year assessment. His life could be said to be very full. There virtually wasn’t a waking moment in which he wasn’t engaging in some passionate learning, whether it be tackling more varying and wackier formations to debug, reading through interesting new textbooks, pondering puzzles in his puzzle picture book, or discovering fascinating details about the world around him. Yet by the end, he only grew less and less satisfied. The more he learned and theorized, the more he realized he didn’t know, and the more engrossed he became in learning new content. It was like diving into a small pool, before realizing that it was actually connected to an enormous ocean. He already couldn’t wait until next year, when he could dive in even more!

In the end, Emmet became too engrossed with his extensive study, and never ended up challenging the more advanced formations in the pile to the right at Maisy’s Debugging Depot. However, he planned to do so as soon as the next year started. Other than that, he became even quicker in debugging the formations in the other piles, although with less radical improvement from before. In the last month, Maisy announced who would be promoted should they choose to continue working at this job the next year, and the first name she mentioned was Emmet. He could only beam proudly and accept the new position. This job was one of the most enjoyable things of his entire life, and he got to get paid for it! It was a total dream!

Finally… The end of year assessment came.

It arrived impendingly and like the apocalypse. The last week, Emmet started to notice everyone around campus, including in his dorm, become increasingly quiet and nervous. This test would rank them within the grade! Even if they were first years, if they were ranked within the lower half, that meant it was likely they could be cut the following year!

The atmosphere turned eerie. Usually boisterous and active, the campus seemed to have settled down, everyone shutting themselves indoors to cram. It was like a great tribulation was coming.

Maisy also became more and more quiet, mentally preparing herself for the trial ahead. The seventh year assessment was a completely different beast from any other year’s. If you made it to the seventh year, that meant you were extremely talented and capable, the best of the best. Thus, the assessment was created to match that, becoming increasingly enigmatic and open-ended. They would cover the highest level contemporary topics, testing the student on their level of engagement with current developments and theories, as well as pound in the fundamentals. It was likely the most difficult academic test throughout the entire formations world. If you didn’t know your fundamentals down pat, then you would be stripped out immediately, as everyone else certainly would, gaining an edge off of you right off the bat.

Emmet could only silently give her encouragement, wishing for her success. He believed that she could do it! She was probably the most amazing formations expert he had ever met. If she couldn’t do it, then who could?

This entire semester, Emmet felt sad that he wasn’t able to talk to Maisy much. Aside from showing her his hoverboard (she loved it, praising him as ‘a total stud’ and calling him her ‘number one disciple’), asking for recommendations for courses, and greeting her when he came in to work, he hardly had the chance to talk. She would always be busy just sitting alone at her desk, completely focused on a scattered pile of texts in front of her, often also simultaneously experimenting with a formations disk in her hands, so he didn’t want to bother her. Sometimes, he would look over her shoulder and just stare in awe at what she was looking at. Complicated formula after convoluted terminology… many of the research papers were like encrypted puzzles in of themselves.

The day of the assessment, it was like doomsday had come. Nearly 14,000 people woke early at 6, solemnly flooding out into the streets from their residences toward designated test halls spread out across the campus. There, they would go through nearly twelve hours of written testing, construction prompts, debugging sheets, and other related tests corresponding to their levels. How well they did would determine many of their fates. The more years that they advanced at the Gemini Academy, the more prestige they would receive within the formations world.

The day passed slowly. Twelve hours later, when it had turned to night, a bell rang, and it was over.

14,000 people flooded out from the test centers, exhaustion on everyone’s faces. Twelve full hours of expensive testing, building constructions, writing down theory, calculating mana efficiencies… it was grueling. At that moment, no one had more of a desire than to just get back to their dorm and pass out.

Maisy walked out of one of those buildings, stroking her chin curiously. Unlike everyone around her, she didn’t seem tired, but rather pensive.

“That question…” She mumbled to herself.

She thought back to something on the assessment she had just took. On one of the written portions of the exam, one of the last prompts had been a theoretical questioning about a famous open problem, called the Kamaran multi-directional mana separation problem. It prompted the student to discuss several possible frameworks in order to approach it and possibly solve it. She understood why it was there - just this past year, there had been a publication from a famous formations expert musing on the problem and suggesting some paths to a solution. Clearly, it was testing whether the student had paid attention and understood the prominent publication.

This Kamaran multi-directional mana separation problem was important, as it was a structure of microcircuit logic that had potential to be used in many different types of formations, and had been pondered about by experts for hundreds of years. If solved, it would increase many formations’ efficiency as well as dramatically decrease their mana costs. Although currently, there was an algorithm proposed that successfully completed the task, it didn’t have a good efficiency, and took exponential more mana-consumption when it was scaled up, rendering it useless. The question was: was there an algorithm that would solve it efficiently, and scale with an efficient polynomial increase in mana cost? - polynomial being an increase rate one dimensional lower than exponential, and generally considered fast and efficient.

This algorithmic speed and mana exertion of microcircuits was an incredibly important consideration for formations - for a small formation, it might not be too noticeable, since there were fewer microcircuits, and the input mana size was small, so it would activate in more or less the same time and with same mana cost. However, for a bigger scale formation, say, a city-wide protection warding formation, it mattered a lot more. If some of the formational microcircuit algorithms implemented in such a formation’s countless microcircuits even had slightly worse than optimal efficiencies and mana costs, then it would waste a tremendous amount of resources as well as take way too long to function properly.

The expert who wrote the paper hadn’t quite solved the open problem, but had laid out several new possible paths to a solution, each not quite reaching an efficient formational microcircuit algorithm but having promising beginnings. Maisy had read it extensively, and felt she had a pretty good understanding of his thoughts.

However, just as she had placed her pen to the paper, she couldn’t help but think back to something that had happened while she had been studying the topic. The cute squire boy Emmet had taken a break from his job and come to peer over her shoulder, staring down at the complicated material. Wanting to tease him, she let him stare for a few minutes, and then suddenly jerked her head back and asked what he thought about it. Taken by surprise, Emmet could only stutter with a red face. However, he eventually did stammer something out in a messy ramble.

However, for some reason, what he had said stuck in Maisy’s mind.

“...uh… ah… um... it’s… I… I think… um, well it’s probably not solvable at all… right? Like, if you think about it… pretty much all of the Bakastarian set can be reduced to it… so if you could solve it, couldn’t you solve pretty much all of the Bakastarian set?”

If you could solve it, couldn’t you solve pretty much all of the Bakastarian set? That question had rung out through Maisy’s head.

There were two complexity classes commonly used to broadly classify every formational microcircuit algorithm: Atistonal, the set of what can be solved in efficient polynomial time, and Bakastarian, the set of whose solution can be verified in polynomial time, but not necessarily solved. Anything in the Atistonal set was also in the Bakastarian set, but only a few problems in the Bakastarian set were in the Atistonal set. Although it was unknown whether the Kamaran multi-directional mana separation problem was contained within the Atistonal set, and was able to be solved, it was definitely contained in the Bakastarian set, famously proven by a personage named Beromi over two hundred years ago.

If Kamaran multi-directional mana separation could really be solved, then that meant it in the Atistonal set as well. The important point was, if a different problem that was also in the Bakastarian set could be reduced to Kamaran multi-directional mana separation, then that meant it could also be placed within the Atistonal set, indicating that it could also be solved! The shocking thing was, this Bakastarian set was full of hundreds of unsolved problems, some more important and famous than even Kamaran multi-directional mana separation. So if these important unsolved problems could be reduced to Kamaran multi-directional mana separation, then that meant they also had solutions!

She began writing down several other big unsolved problems within the Bakastaran set. Nearly an hour later, she shockingly discovered that over five of them were able to be quickly reduced to Kamaran multi-directional mana separation! Emmet was right! She didn’t have time, but she was confident that she could find many more that would reduce to it. If Kamaran multi-directional mana separation could be solved, then likely tens of other unsolved problems would also instantly become solved! Thus, it was it was unlikely that there would be a solution!

After that, the ideas flowed and maisy wrote down page after page of reductions and corresponding logic.

Thinking about it now, Maisy rubbed her face. She didn’t know why she had written all of that down in the moment. She was completely unsure of her answer, she had just been for some reason madly inspired by something Emmet had said. It would probably have been better to just go with the safe option and describe out what that one expert had said in his publication.