Chapter 94: New Problem

Chapter 94: New Problem

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 dont respond, its that Im not unavailable. To leave the array, just walk in a random direction. It shouldnt 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?

Thats 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 didnt tell anyone!

Haha, dont worry I trust my brothers! This isnt 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 shouldnt 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. Wont 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 wasnt hallucinating. For a moment, he wanted to believe he was just in other of Han Lingyuns 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 hesitated for a good minute, with the book still opened at its 1stpage.

I cant hold back from diving into that manual! I dont think I would be able to cultivate well with this desire gnawing at me anyway. Okay Ill just look at one problem. Ill work on it while Im 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 dont 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 Wens explanation. He couldnt wait to start searching. This type of problem found an echo in the younger years of his previous lifes 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 to test different ideas, including some paths that didnt work out. It wasnt 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 couldnt help but marvel. Indeed, this could simplify a line of runes by a considerable amount! Im 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. Im 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 wouldnt 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!