Han Lingyun had ignored the alarms that the trespassers had set off while he was cultivating but their presence had still intrigued him.

This evening, when Shen Lin entered his formations in turn, Han Lingyun deactivated some to let him through. He had given specific instructions to his 3 best friends so that he could tell when they were the ones coming.

It went along the lines of “Walk in the direction of that rock for 20 steps, close your eyes and back up 10 steps. Then open your eyes again and advance 30 steps towards this tree. Afterwards, I will open the bewildering formation so that you can enter. If I don’t respond, it’s that I’m unavailable. To leave the array, just walk in a random direction. It shouldn’t take more than a minute.”

Thus, Han Lingyun recognized that one of his friends was here to find him.

“Brother Shen! Are you here because of the disciples that came today?”

“That’s right brother Lingyun. This week, word that you lost your cultivation spread like wildfire and some wanted to find you to verify it. I can say with absolute certainty that brother Yang and I didn’t tell anyone!”

“Haha, don’t worry I trust my brothers! This isn’t too unexpected by the way. I was able to fool the disciples, but I cannot fool the Elders. With their spirit sense and a cultivation at the Core Formation realm, those that were present at the ranking competition should have all noticed immediately my true cultivation.”

Han Lingyun added internally. ‘It seems that when Elder Wen was at the school, some were keeping it to themselves to avoid potentially angering him. It shouldn’t be a coincidence that word spread just after Elder Wen left. I guess other disciples are also starting to have special relationships with some Elders.’

Shen Lin looked concerned. “Won’t this be a problem, brother Lingyun?”

“Not at all, I might actually go out and disclaim the rumor.”

“What are you talking about brother? It would be awkward with some of the qi condensation cultivators. You cannot lie about your cultivation too, some will inevitably---”. Shen Lin stopped on his tracks and his jaw dropped.

Han Lingyun was smirking, a ball of black qi dancing in his hands.

“Am I not at the qi condensation realm?”

Shen Lin pinched himself to verify he wasn’t hallucinating. For a moment, he wanted to believe he was just in other of Han Lingyun’s illusions but his friend had never been able to create dynamic images like that. “H-How? It has not even been a month!”

Han Lingyun’s smirk just widened. “Hop on your flying treasure and take my hand, I’ll lead you out.”

Disciples were stunned to see Han Lingyun speeding in the air on his flying sword. He came to a popular training ground and hovered a dozen of meters in the air.

Just as some disciples were recovering from their surprise, Han Lingyun slashed a couple of times with his sword towards the ground and disappeared in a cloud of black qi.

After a couple of breaths when the smoke had dissipated, Han Lingyun was nowhere to be seen. On the ground, the stone floor had some profound cuts left by his invisible qi blades. Together, the cuts read. “Don’t worry about me.”

With this, the rumor evaporated even faster than it began. Many began criticizing those that had been the most vocal or adamant. Even most of the Elders disregarded the idea that Han Lingyun had lost his cultivation. Perhaps he was just under a peculiar state during the competition due to a special type of training or pill being consumed. This wasn’t unheard of. In fact, there were even specific pills with the only objective of hiding or locking one’s cultivation. Han Lingyun didn’t seem to have any reason for having consumed one though.

A good portion of the disciples actually were relieved that their number one disciple was fine. After Yu Tao and Wei Muchen’s disappearance, some were beginning to worry all their best elements would leave or have problems. A strong number 1 was better for the morale of the promotion.

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But those that had received the greatest shock after Han Lingyun revealed he was at the qi condensation stage were none other than Yang Haoyu and Wang Yimo. Unlike the other disciples who quickly believed it was all baseless speculations, they knew from Han Lingyun himself that he had to recultivate from scratch. And in less than one month, he was already back at the qi condensation stage!

After his little stunt, Han Lingyun came back to his cave at the top of the 3rd mountain. His re cultivation to the early qi condensation realm has been smooth sailing. He even felt slightly stronger than previously. Most importantly, losing his cultivation hadn’t affected the physical advantages he had previously gained, such as the strengthening of his tempered bones and an increased night vision.

After settling his thoughts, Han Lingyun’s gaze inevitably shifted to the books Elder Wen had left behind. He had yet to touch them after dedicating himself whole heartedly to cultivation for the past week. Han Lingyun suddenly became emotional. His teacher had been a silent protector all this time and now he had to fend for himself. His absence was already being felt with the little events like this rumor. The fact the rumor was true wasn’t the issue, it would probably never have spread in the first place with Elder Wen at the sect. Thinking back to his teacher, Han Lingyun picked up the pile of books he had left him and skimmed through their 1st page and contents.

Han Lingyun’s raised his eyebrows after seeing the last manual titled A gift for my brilliant disciple. Han Lingyun immediately opened to the 1stpage to find a small note from Elder Wen.

“My dear disciple, I know that you still love some fun math problems. Ever since we had our bet 2 years ago, my perspective on its usefulness in the array field has shifted greatly.

I have thus tried to find more mathematical results that have applications in array formations. In this book, I have collected some of the most interesting ones I found, with their applications. If you can’t find the solution, you won’t find the answer in this book, but some hints are at the end of the book. The level of the problems varies greatly. You should find a couple easy ones but there might even be one or two problems that I have yet to solve and I’m not telling you which one’s which. Good Luck!

I also slipped in there a couple of fun problems that don’t necessarily have a link with the field of arrays, but I know that you will enjoy them.

Your grateful teacher,

Wen Chen.”

It was only a week since Elder Wen had left but Han Lingyun had a wide smile with teary eyes after reading this letter. He realized that he had truly been lucky to be examined by Elder Wen before even becoming an official disciple.

Han Lingyun hesitated for a good minute, with the book still opened at its 1stpage.

‘I can’t hold back from diving into that manual! I don’t think I would be able to cultivate well with this desire gnawing at me anyway. Okay I’ll just look at one problem. I’ll work on it while I’m taking pauses in my cultivation.’

Han Lingyun turned to the second page.

Problem 1:

Find all functions such that for all x and y in f(xf(x)+f(y))=f(x)^2 +y

Motivation:

It is often complex to understand how qi will behave after inscribing a new rune to an already existing runic phrase. Because the runes don’t have independent relationships, one needs to factor the combination of runes and feedback loops to predict the result.

On some occasions, simplified math equivalents can be found with functional equations, with x and y being two rune segments and ‘f’ representing the effect of the runes on qi and of their interaction. I stumbled on this equation once and the result helped me simplify a line of runes in a sect array.

Han Lingyun had a wide grin on his face as he read the problem with Elder Wen’s explanation. He couldn’t wait to start searching. This type of problem found an echo in the younger years of his previous life’s memories, when he participated in various math competitions. He still had some of the good tricks in mind to solve these questions.

First plug in 0 for x or for y, and try simplifications. Because this path looked promising, Han Lingyun looked for possible values of t such that f(t)= 0. He found that there could only be t=0 if such a t existed. It was easy to find one such value with the initial equation, which also gave f(0)=0 as a bonus.

After playing a bit further, he then found that f was an involution and managed to find a simpler form for f by plugging in f(x) in place of x in the initial equation. He found the simplified form: f(x)^2=x^2.

Finally, after an additional step on the sign of ‘f’ on the original equation and a quick verification, Han Lingyun was able to show that:

Either f(x)=x for all x, or f(x)=-x for all x.

Han Lingyun swiped his forehead after finding the nice answer. The process of solving may have sounded simple but it required him to test different ideas, including some paths that didn’t work out. It wasn’t too complicated, but one always felt a moment of pride when completing a problem.

As Han Lingyun gaze went back to the simple solutions in comparison to this ugly looking functional equation, he couldn’t help but marvel. ‘Indeed, this could simplify a line of runes by a considerable amount! I’m so excited to search for all the problems Elder Wen provided me. This manual is another goldmine! He truly is a generous and caring teacher. I’m glad I was able to direct him in a direction that made him take a step forward in his field.

After another short recollection, Han Lingyun put the math problem book to the side and sat down for his next cultivation session. He wouldn’t mind alternating between cultivation, math problems, some quick exercises in front of his cave and a bit of sleep for the next few weeks and that was exactly what Han Lingyun was planning on doing!