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 so 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 ca today?
Thats right brother Lingyun. This week, word that you lost your cultivation spread like wildfire and so 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 imdiately my true cultivation.
Han Lingyun added internally. It seems that when Elder Wen was at the school, so 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 so 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 so of the qi condensation cultivators. You cannot lie about your cultivation too, so 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 mont, 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 Lingyuns smirk just widened. Hop on your flying treasure and take my hand, Ill lead you for the way out.
Disciples were stunned to see Han Lingyun speeding in the air on his flying sword. He ca to a popular training ground and hovered a dozen of ters in the air.
Just as so disciples were recovering from their surprise, Han Lingyun slashed a couple of tis 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 so profound cuts left by his invisible qi blades. Together, the cuts read. Dont worry about .
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 consud. This wasnt unheard off. In fact, there was even specific pills with the only objective of hiding or locking ones cultivation. Han Lingyun didnt seem to have any reason for having consud one though.
A good portion of the disciples actually were relieved that their number one disciple was fine. After Yu Tao and Wei Muchens disappearance, so were beginning to worry all their best elents would leave or have problems. A strong number 1 was better for the moral of the promotion.
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 ca 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 hadnt 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 Lingyuns gaze inevitably shifted to the books Elder Wen had left behind. He had yet to touch them after dedicating whole heartily to cultivation for the past week. Han Lingyun suddenly beca emotional. His teacher had been a silent protector all this ti 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 wasnt 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 skimd through their 1st page and contents.
Han Lingyuns raised his eyebrows after seeing the last manual titled A gift for my brilliant disciple. Han Lingyun imdiately opened to the 1stpage to find a small note from Elder Wen.
My dear disciple, I know that you still love so 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 so of the most interesting ones I found, with their applications. If you cant find the solution, you wont find the answer in this book, but so 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 Im not telling you which ones which. Good Luck!
I also slipped in there a couple of fun problems that dont 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 cant hold back from diving into that manual! I dont think I would be able to cultivate well with this desire gnawing at 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 so occasions, simplified math equivalents can be found with functional equations, with x and y being two rune segnts and f representing the effect of the runes on qi and of their interaction. I stumbled on this equation once and the result helped 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 mories, when he participated in various math competitions. He still had so 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 so paths that didnt work out. It wasnt too complicated, but one always felt a mont 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 . 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, so 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!
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