Chapter 1 "Wait"

"Hold on," Hornheim reacted instantly, countering, "That doesn't sound very fair. The coin is yours, and you're the one flipping it. How can I be sure the game is fair?"

"If I intended to cheat you, I wouldn't have appeared in your sight." Claude pointed towards the corpse of the giant boar below the stone pillar, saying casually, "See that boar? There are countless creatures in this forest that are many times stronger than it. Even if you're well-prepared, if you can't find the pattern in which the forest automatically changes its position, you'll be exhausted to death in this woodland. And all I need to do is wait quietly in the distance to collect the supplies and items on your bodies."

Claude nonchalantly tossed the coin in his hand towards Hornheim. The coin flew halfway through the air before being seized by a transparent Mage Hand released by Hornheim, suspended in mid-air.

"Don't you have a magic book?" Claude pointed to the Sand Book in Hornheim's hand, saying, "You can use it to test the coin and see if I'm cheating."

He actually agreed to let me use magic? Hornheim narrowed his eyes slightly. The Sand Book turning in his hand paused almost imperceptibly, then quickly locked onto several pages, releasing several different identification spells.

This coin is made of a brass alloy, with a uniform thickness and smoothness. The front has a worn human head, and the back has a snow mountain. It is 40 millimeters in diameter and weighs more than a real-world coin. It's a bit like a special commemorative coin.

No supernatural traces detected... Hornheim silently reduced the magical power output of the detection spell.

Claude took the coin from the Mage Hand, saying, "Heads for the head, tails for the snow mountain. To be fair, I'll go first in the first round. The sequence I choose is tails-heads-tails. That is, snow mountain - head - snow mountain."

"..." Hornheim thought for a moment, "Then I choose, tails-tails-heads."

"Oh? Quick reaction." Claude grinned. Hornheim smiled faintly, not explaining.

"I'm starting to toss it." The arm in the middle of Claude's chest used its thumbnail to hold the coin and flicked it upwards. The coin rose high, spinning in the air, reflecting the sunlight that pierced through the dense canopy.

Hornheim's pupils suddenly contracted. The detection spell released from the Sand Book in his hand frantically calculated the coin's kinetic and potential energy conversion trends, mapping and estimating the trajectory of the coin's rotation, flight, and fall.

In the short time Claude explained the rules, he had already understood the game's pattern.

[Both sides choose a sequence of three coin tosses, heads or tails, and keep tossing the coin until the sequence appears, which counts as a win for that round.]

This rule seems fair, but it is actually unequal. The side that chooses the sequence later has a huge advantage. No matter what sequence the first player chooses, the second player can have a relatively better solution.

Taking Claude's choice of tails-heads-tails, or 010, as an example, Hornheim's choice of tails-tails-heads, or 001, is probabilistically superior to the former.

Because in an infinitely long sequence (assuming the coin is being tossed continuously), the winning condition for 001 is the appearance of 1001, or the appearance of 0001.

To win, 010 can only have the possibility of 1010.

Because if the sequence 0010 appears, then the second player, who chose 001, will win directly before 010.

It is equivalent to one less winning possibility than the opponent.

From Hornheim's point of view of choosing 001, one of the prerequisites for the opponent to win is actually the basis for one's own victory.

Unless the result is directly thrown in the first three throws, otherwise the probability of 001 will always be prioritized over 010 in an infinitely extended ray-shaped sequence - because the game rule is [the corresponding sequence appears first], rather than [find it in an infinitely long sequence].

* * *

Clap.

Claude caught the coin with the back of his left hand, covered it with his right palm, and then slowly opened it. The side of the coin facing up was the human head, which is heads.

"Ah, no one got it on the first toss." Claude grinned, "Let's do it again."

He continued to toss the coin.

Second toss, heads.

Third toss, tails.

The first three throws were heads-heads-tails. No one guessed correctly.

Hornheim was silent. The probability of hitting all in the first three throws was too low. The next few throws are the beginning of the game.

Fourth throw, heads.

Fifth throw, heads.

Sixth throw, tails.

Seventh throw, tails.

Eighth throw... heads.

Won.

A flash of light flashed deep in Hornheim's eyes. Claude sighed, looking at him and smiling, "Good luck. You won the first round."

Hornheim narrowed his eyes. It wasn't luck, but pure probability calculation. His 001 sequence had two-thirds, or about 67%, chance of beating 010.

This is not to say that 001 is the optimal solution. As he calculated, among the eight possibilities of the three-digit sequence of 0 and 1 (000, 010, 001, 100, 110, 111, 011, 101), the winning probability of 111 and 000 is the lowest. 011 and 100 both have seven-eighths, or 87.5%, probability of winning.

The sequences 100, 001, 011, 110, etc., are mutually superior and inferior solutions, forming a cycle of mutual generation and mutual restraint, like rock paper scissors.

For a gambler, a 55% chance of winning is already a huge advantage. If he can control the odds of winning at 67% or even 75%, then he can almost bet all his wealth, and there is no reason not to continue to bet.

Claude said slowly, "Best of five games. In the second round, you choose first."

"..." Hornheim thought for a moment and said slowly, "I choose, heads-tails-tails."

Although there will be cycles between the sequences, the probability of 011 and 100 being restrained is relatively low.

Claude smiled, "Then I choose, heads-heads-tails."

As expected, Hornheim's heart sank. The sequence chosen by the other party was probabilistically superior to his first-mover solution, but... there may not be a chance to make a comeback.

The result of the second round of throws was: tails-tails-heads-tails-heads-tails-tails-tails.

Won again.

Hornheim's furrowed brow eased slightly. The only solution for his heads-tails-tails to win is to make a double judgment in addition to the first three throws, and the probability of winning is not high.

"Wow, it seems luck is really on your side." Claude whistled, tilting his head, with a strange arc rising at the corner of his mouth, "It's already match point. So in the third round, I choose, heads-heads-heads."

"Hmm?" Hornheim's eyes narrowed, and he glanced at the other party deeply. The probability of heads appearing three times in a row was not high. If he chose tails-heads-heads to suppress it, then the other party's winning rate would be infinitely reduced.

He must directly throw the result of heads-heads-heads in the first three throws to win - because once the sequence exceeds the first three digits, then there will inevitably be a situation where [there must be tails before heads-heads-heads], no matter what, the party that chooses tails-heads-heads will not lose.

The probability of winning reaches... 87.5%.

"I choose, tails-heads-heads."

Under the cold probability calculation, Hornheim had no reason to choose other solutions and immediately gave the answer.

"Very good, then, I'll start." Claude's thumbnail flicked up the coin.

The circular slice of brass alloy rotated in the air, seemingly slowly but actually quickly, and finally fell on the back of Claude's hand, where it was quickly caught.

The result showed... [heads]

"The first throw is not bad for me." Claude grinned, ignoring the dense monitoring techniques in the void like probes, and placed the coin back on his thumbnail, and flicked it up again.

*clang*

Still, [heads].

The aura around Hornheim suddenly changed. His 87.5% winning rate was based on the fact that the result of heads-heads-heads could not be thrown in the first three rounds.

For the other party to win means the probability is one-half multiplied by one-half multiplied by one-half, for a total of only one-eighth possibility.

"The third round..." Claude flipped the coin skillfully, and tossed it again.

The brass alloy coin flew, fell, and was clasped in his palm with a *thud*.

[Heads].

"With a one-eighth probability, I won." Claude whistled easily, "The fourth round..."

"Wait a minute." Hornheim raised his palm, "How about I toss the next two rounds?"

He has never been a superstitious person when it comes to probability. As a player, he also believes that the system will not set up a kind and benevolent old man full of magic weapons in a difficult copy as a caring guide for players.

The other party must have used some means to make him fall into a trap...

"Okay."

Unexpectedly, Claude nodded generously and threw the coin to Hornheim casually.

As before, Hornheim grabbed the coin with an invisible Mage Hand and examined it more carefully with detection magic.

The human head on the coin was severely worn, and one could vaguely see that its face had distinct edges and corners, deep eye sockets, a straight nose bridge, and regular facial features, with short curly hair.

From the perspective of universal aesthetics, he is considered a handsome man, which is very different from the three-armed monster Claude in front of him.

Hornheim raised his head and asked, "Who is the head on this coin?"

Since the coin was severely worn, he didn't have to worry about the situation where [the person on the coin is extremely well-known in the script world, and they don't know each other, which would make Claude suspicious of their identity]. Anyway, he could attribute the reason to "unrecognizable."

"Who else could it be? Ti Xiu, the head of the Seven Pirate Kings of the Boundless Deep Sea, the commander of the Invincible Fleet, and the symbol of all the wealth in the world." Claude was a little confused, "I accidentally picked up this copper coin in this forest. It is estimated that Ti Xiu and his subordinates dropped it when they came to the Holy Mountain to worship a long time ago, or it was brought in by a traveler from a certain generation."

He glanced at Hornheim strangely, "Could it be that you thought the person on the head was me? Am I that ugly?"

"Uh, no, no." Hornheim resisted the urge to complain. The strangeness of the three-armed monster's appearance is enough to throw it into a horror game's sinister scene in the real world, to be a monster that acts as a jumpscare (sudden scare) program effect. It's one in the sky and one underground compared to the handsome guy on the coin.

It's about the difference between Duan Yanqing and Duan Yu, right... But it may be normal. Anything can happen in the killing game. Maybe Claude's race has abnormal aesthetics. Before, a slightly fat player entered the Underdark in the script world and was warmly received by a Drow elf clan that regarded fat as beautiful. They killed the thin and handsome guy in the player team, leaving only him, and lived a happy life ever after - although the result was that this group of players fell into the web of cannibalistic spiders and everything was an illusion, it does not affect the effectiveness of this rule.

Don't easily discuss beauty or ugliness with the natives of the script world.

Claude spread his hands and said, "I chose first in the last round. You go first in this round."

Hornheim took a breath, "I choose, heads-tails-tails." Just an accident. Before he sees the problem, the probability of superiority and inferiority restraint still takes effect.

Claude grinned, "Then, I choose tails-tails-tails."

Again! Hornheim's heart sank. The probability of heads-tails-tails is also superior to tails-tails-tails, and the winning rate is still 87.5%.

In this round, he was the first to choose. After he chose heads-tails-tails, the other party could completely choose heads-heads-tails to restrain it with a 67% winning probability, but the other party did not choose heads-heads-tails.

Instead, he chose tails-tails-tails, which has a winning rate of only one-eighth. There must be something wrong...

"Since you've decided, let's start tossing." Claude glanced at the coin floating in the air, tilted his head, and urged.

"...Okay." Hornheim narrowed his eyes. The coin was firmly grasped by his Mage Hand, and the power to toss was in his hands.

The Mage Hand is an extension of his own magical power, and it is even more flexible and controllable than a hand.

'Let me see if you are cheating...' The invisible palm flicked the coin, and the coin flew and rotated by itself, drawing a straight trajectory in the air.

*Thud*.

The Mage Hand, completely composed of magic, firmly caught the brass alloy coin at the same height as the toss.

A gleam flashed deep in Hornheim's eyes. He didn't cheat, but he already knew the result.

Those monitoring magic techniques infused with magic in the air were intently staring at every rotation of the coin, and had already calculated the result before it fell and was caught.

I've already won. Hornheim felt the magic power draining from his body (the surrounding low ionization energy environment is extremely unfriendly to players specializing in spellcasting), and slowly uncovered the coin.

Wait.

Snow mountain?

Hornheim's eyes widened slightly, and he suddenly raised his head to stare at Claude, his voice low, "You cheated."

"Did I?" On the stone platform, Claude smiled, "You tossed the coin, and you caught it. As a mage, you should be very clear about whether I interfered in the tossing process just now. Don't make the situation too ugly just because you are going to lose..."

The three-armed monster scratched his cheek lightly with his right hand, drawing streaks of blood, and smiled, "Besides, even if I cheated, as long as you don't have evidence to prove it, the game is still valid."