Ten thousand I am across the heavens

Everyone looked at the question, then seemed to ponder it. The question wasn't complicated; in fact, it was quite simple. Everyone could understand it, and there were certainly no complex calculations involved. Yet, everyone was caught up in deep thought.

The premise is simple: there are 100 people on a small island. Each of them has blue eyes, but they are unaware of this, and no one is allowed to discuss the color of their eyes with each other.

They could only see that everyone else had blue eyes. They didn't know what color their own eyes were. And there was a rule on the island: anyone who knew their own eye color had to jump into the sea that very night.

Everyone on the island is smart enough and will absolutely follow the rules mentioned above.

For the first few days, these people lived in peace. No one jumped into the sea, until an outsider suddenly arrived and told them, "At least one of you has blue eyes."

What will happen next? Will someone jump into the sea? When will they jump? Why?
Black City pondered the question: "The answer should be that nothing will happen, right? Wasn't what the outsider said pointless? If I were on this island, I could see 99 pairs of blue eyes every day. Then everyone else could see at least 98 pairs, so everyone should know that at least one person on this island has blue eyes. He said something pointless, it shouldn't have any impact. These people were fine before, and they should be fine now."

Everyone else nodded; this explanation seemed the most reasonable. No matter how you looked at it, what the newcomer said was pointless. A pointless statement shouldn't have any impact.

However, the screen quickly displayed "Incorrect Answer." At the same time, everyone could feel the barrier's strength slightly increase. Although not by much, it was clear that an incorrect answer would amplify the power of this deadly attack. Therefore, it was impossible to guess the answer repeatedly. Even if they guessed correctly, they had to explain why. Otherwise, after guessing wrong many times, the barrier would strengthen, making it impossible to escape.

"How could this be? This answer is wrong. I just can't understand it. Could it be that knowing someone has blue eyes really changes something?" Even though Black City had split into so many clones, each capable of independent thought, he still couldn't figure it out. He felt like he was overlooking something, yet his previous logic seemed flawless.

The others were also thinking. Chen Feng scratched his head and said, "Although I don't know the answer, I feel it's very likely that on the 100th day, everyone jumps into the sea together. I'm just guessing. But my intuition is this: on the first day, no one jumps into the sea. Everyone knows there are at least two blue-eyed people. On the second day, no one jumps into the sea, so everyone knows there are at least three. By the 99th day, no one has jumped into the sea, so there are at least one hundred. Then everyone knows that everyone has blue eyes. And then they all jump together."

I haven't quite grasped the logic here, but it just feels like that to me. Anyway, guessing once won't significantly increase the barrier's power. I might as well guess again; even if I'm wrong, it won't matter.

"Hmm... my intuition also tells me something bad will happen on the 100th day. Although I think your reasoning has a lot of problems—everyone knew from the beginning that there were at least 99 pairs of blue eyes, so it shouldn't be possible to conclude that there are at least two pairs of blue eyes on the first day—I think your guess is very likely correct. Anyway, if the answer is correct, there should be feedback. Even if the given principle is wrong, it can at least prove that the answer is correct. If you work backward from the answer to deduce the principle, the difficulty should be reduced a lot."

I really think what the other person said makes a lot of sense. The question doesn't seem difficult, it's just very convoluted. You might actually guess the right answer based on your intuition.

So he immediately stepped forward and answered on the screen: "The answer is that on the night of the 100th day, everyone jumped into the sea together."

"Correct answer. Please explain why everyone jumped into the sea on this day?"

Hei Cheng sensed that the overall strength of the barrier had decreased by about ten percent, indicating that his previous answer had indeed been a lucky guess. He then repeated the reasoning he had just heard. Unsurprisingly, it was judged as an incorrect answer; his answer, which he could find flaws in, was definitely problematic. As the answer was incorrect, the barrier's strength seemed to return to normal, though overall it was weaker than before. This proved that answering the question here was worthwhile. Now, even if he forcibly broke the barrier, the cost would be less. The next step was to figure out why the answer was as it was. It seemed that guessing the correct answer wasn't difficult. But to understand why? A thorough discussion was necessary.

However, after much discussion, the group couldn't reach a conclusion. They split into two factions. One faction fundamentally disagreed with the answer, believing they had fallen into a trap. They argued that the answer of everyone jumping into the sea on the 100th day was inherently flawed because everyone knew from the beginning that there was at least one pair of blue-eyed people on the island, and that everyone else knew this too. Therefore, no matter how they reasoned, no one would die suddenly. It was all a trap set by the cunning people in the underground base.

The other side had its own reasoning. Although it differed slightly from Chen Feng's previous reasoning, it was more refined and sounded more logical, but it still seemed to have flaws.

They believed it was reasonable to assume that if only one person among these people had blue eyes, then after learning on the first day that there was at least one pair of blue eyes, that person would know it must be them. Therefore, they would have no choice but to jump into the sea on the first night.

In a scenario where there are two pairs of blue-eyed people on the island, each person would perceive only one other person as having blue eyes. Since the first day was Christmas Eve, they discovered that neither of them had jumped into the sea, and thus realized they also had blue eyes. So they both jumped.

When there are three pairs of blue eyes on the island, and each of the three people only sees two pairs of blue eyes, nothing happens on the first day. On the second day, the two people who jumped together didn't, which means that they also see more than one pair of blue eyes. This means they are the ones with the blue eyes, so on the third day, all three will jump together. This continues until everyone has blue eyes—that is, on the 100th day, everyone will jump.

This reasoning seems perfectly reasonable, but one reason it's questioned is that it starts with the premise that there's only one pair of blue eyes on the island. However, everyone knows from the beginning that there are at least ninety-nine pairs of blue eyes on the island, and that in the eyes of others, there are at least ninety-eight pairs. It seems that the first step of the reasoning is invalid, so the two sides argue endlessly and can't reach a conclusion for a while.

After discussing for a long time, someone suddenly suggested, "Our brains might not be up to par. But we have a lot of people, so why don't we try a simpler method? Gathering 100 people shouldn't be difficult for us. Gathering 100 identical people shouldn't be difficult either. We can just simulate it as much as possible, right? Anyway, everyone's thinking speed is not slow, so it shouldn't waste too much time." This barrier has trapped them for quite some time. Thinking of this, Hei Cheng nodded in agreement. Although this method was a bit clumsy and would waste some time, it did seem like it might reveal the answer directly. The longer it dragged on, the more things could go wrong. Outside, the eldest son's clones were already engaged in fierce combat. They needed to find a way to break through as soon as possible to go out and help.

"Let's do it this way. Let's assume there's only one person on the island. This person has blue eyes. What would happen if an outsider told him he had blue eyes?"

A man emerged from the Black City. He stood in the center of the barrier. Without hesitation, he answered, "I will know that blue-eyed one is me that very day. And that very night I will have to jump into the sea."

"That's right, now we'll increase the number to two."

So the two Black Cities stood in the center of the barrier.

The first Black City said, "I can only see that your eyes are blue. If I don't want to jump into the sea, I have to believe that my eyes aren't blue, so I have to believe that you can't see any blue-eyed people. Therefore, I hope you jump into the sea tonight. Otherwise, I will die tomorrow."

The second person nodded and said, "That's what I thought too, so the answer is obvious. Neither of us jumped into the sea on the first day. On the second day, we both knew we had blue eyes. So we had to jump together on the second night."

"Okay, let's see what the situation is with these three people now?"

The three black-clad figures arrived at the center of the square. The first one, seemingly lost in thought, said, "Now I have two blue eyes in my eyes, so I definitely don't want to jump into the sea. Therefore, regardless of whether I also have blue eyes or not, I have to believe that I am not blue."

Then I have to believe that these two people only have one blue eye. If that's the case, then they should be fine on the first day. They should both jump into the sea on the second night. If they don't jump by the second night, then my assumption is wrong, and I'll have to jump with them on the third day. Because my assumption is that I don't have blue eyes.”

The second and third Black City nodded in agreement: "We definitely have the same idea. So we won't jump on the first and second days. On the third day, we'll see that everything is still peaceful, and we'll come to the same conclusion as him. Then the three of us will have no choice but to jump into the sea together on the third day."

"It seems there's a pattern. If we continue this reasoning, let's try the fourth day. The fourth day seems to be a crucial point in time. Because on the fourth day, everyone can be certain that others can see at least two pairs of blue eyes. In this case, the information given by the outsider seems to become meaningless, just like the situation when there are 100 people. The outsider's seemingly meaningless statement doesn't seem to provide any additional information."

Everyone had similar thoughts, so the four of them began their reasoning. The first one, Hei Cheng, started reasoning from his own perspective: "It's the same as when there were three people. If I don't want to die, I have to believe that I don't have blue eyes, and right now I can see that three people have blue eyes."

At this point, I have to put myself in Character 2's shoes. Character 2 definitely doesn't want to die either, so he must also believe that he doesn't have blue eyes, even though I can see that his eyes are blue. He will definitely believe that he doesn't have blue eyes, so in his eyes, there can only be two pairs of blue eyes.

At this point, Number 2 has to wait. From my perspective, Number 2 only sees two blue-eyed people. Based on the reasoning above, these two should both jump into the sea on the second day. However, I know they definitely won't jump on the second day, so Number 2's assumption that they don't have blue eyes is invalidated. Therefore, Number 2 must jump into the sea with those two on the third day. If they don't jump either, it proves my assumption is wrong, proving that my eyes are also blue. So, on the fourth day, I also have to jump in with them.

At this moment, Numbers 2, 3, and 4 from Black City were also deep in thought, considering the situation from the outsider's perspective. What the outsider said was still a pointless statement, but following the previous logic, it was indeed possible to deduce the outcome of everyone jumping into the sea. They exchanged ideas and realized that even if everyone shared Number 1's thoughts, the ending wouldn't change.

After reasoning further, they came to a strange conclusion: "That is to say, no matter how many pairs of blue eyes we can see, the previous Christmas Eves are meaningless. When we can see 99 pairs of blue eyes, only the 99th day is meaningful. On the 99th day, if everyone except me does not jump into the sea, then it proves that I and everyone else will jump into the sea together on the next day."

This reasoning is much more complete than the previous one. Although we still can't figure out what the extra piece of information is, we can at least confirm that the outsider did indeed provide additional information. Even if this information becomes almost imperceptible through repeated transmissions in the logical chain, it still exists. Let's try it this time. Using the pattern we summarized from our previous reasoning as the answer, even if it's wrong this time, it shouldn't be too far off. Black City and his team's personal experience convinced most people, because they didn't just test it with 4 people. 5 people, 6 people, 7 people. They tested it with all of them. When they had 7 people, their minds couldn't keep up. Because there were 7 nested layers of logic. Even if they were immortals, whose minds worked faster than ordinary people, forced reasoning almost made their brains burn. But the conclusion they reached still conformed to the pattern they summarized before. It basically proved that the pattern was correct.

The various legions waited outside for the moment the barrier would shatter, and they could also witness the process of those inside answering questions. The commanders of each legion could only sigh helplessly. Although the enemy's cultural level was not high, they were not fools either, and the barrier seemed to be broken a little faster than they had expected.

[To be continued]

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