"Wait,"
Hoenhem instantly reacted and asked back, "Doesn't this seem a bit unfair?
The coin is yours, and you're the one flipping it. How can I be sure of the match's fairness?"
"If I intended to harm you, I wouldn't have appeared in your sight."
Claude casually pointed to the corpse of a giant boar beneath the stone pillar and said, "See that boar? There are countless creatures in this forest that are several tis stronger than it.
Even if you're well-prepared, without discovering the pattern of the forest's automatic position changes, you'll be worn out and perish here.
And all I need to do is quietly wait in the distance to harvest the materials and tools from your bodies."
Claude nonchalantly tossed the coin in his hand to Hoenhem,
The coin was caught mid-air by an invisible Mage's Hand released by Hoenhem, hovering there.
"Don't you have a magic book?"
Claude pointed to the Book of Sand in Hoenhem's hand and said, "You can use it to check the coin and see if I'm cheating."
Allowing
to use spells?
Hoenhem squinted slightly, the Book of Sand in his hand paused almost imperceptibly, then he quickly locked onto a few pages and cast several different identification spells.
The coin is made of brass alloy, evenly smooth in thickness, with a worn human face on one side and a snow mountain on the other, a diater of 40mm, and heavier than a coin from the real world, resembling a special commorative coin.
No supernatural traces detected...
Hoenhem silently reduced the mana output for the detection spell,
Claude took the coin back from the Mage's Hand and said, "Face is the head, snow mountain is the tail. To show fairness, I'll start the first round.
My chosen sequence is tail, head, tail.
That is, snow mountain — face — snow mountain."
"..."
Hoenhem pondered for a mont, "Then I'll choose tail, tail, head."
"Oh? Quick response."
Claude grinned, and Hoenhem smiled faintly without explaining further.
"I'll start flipping."
Claude used his thumb to flick the coin in his chest, sending it upward.
The coin rose high, spinning continuously in the air, reflecting sunlight through the dense treetop canopy.
Hoenhem's pupils contracted sharply, and the detection spells from the Book of Sand in his hand frantically calculated the conversion trend of the coin's kinetic and potential energy, mapping and predicting its rotational path, projection, and fall.
In the brief mont Claude explained the rules, he already understood the ga's pattern.
[Both sides choose a sequence of three flips, keep tossing the coin until the sequence appears, which counts as a victory round.]
This rule seems fair but is not equitable. The party choosing the sequence second has a significant advantage, regardless of the initial sequence chosen by the first party; the second party has a relatively better solution.
Take Claude's chosen sequence tail-head-tail, that is, 010 for example,
Hoenhem's choice of tail-tail-head, that is, 001, is more probable.
Because in an infinitely long (assuming the coin keeps being tossed) sequence, the winning conditions for 001 include 1001 and can also be 0001,
Whereas for 010 to win, it can only be 1010,
If a 0010 sequence appears,
The party that chose 001 will win before the one with 010.
Which ans one less chance for the opponent to win.
In terms of probability, the 001 sequence benefits from the foundational pre-condition that also leads to the opponent's win.
Unless the results appear directly within the first three flips, 001 will always outperform 010 in an endlessly extended ray-like sequence — because the ga rule is [first to appear], not [find in an exhaustively long sequence].
Pa.
Claude caught the coin with the back of his left hand, covered it with his right palm, and then slowly revealed it.
The top side of the coin showed the face, aning heads.
"Ah, nobody won on the first toss."
Claude grinned, "Let's go again."
He continued flipping the coin.
On the second toss, heads,
On the third toss, tails.
The first three tosses were heads-heads-tails, and no one guessed correctly.
Hoenhem remained silent because the probability of hitting all three in the first tosses is too low; the subsequent ones are where the ga starts.
On the fourth toss, heads,
On the fifth toss, heads,
On the sixth toss, tails,
On the seventh toss, tails,
On the eighth toss... heads.
Won.
A gleam flashed deep in Hoenhem's eyes,
Claude sighed, looked at him, and said with a smile, "Lucky you. You won the first round."
Hoenhem squinted slightly; it wasn't luck, it was pure probability calculation. His sequence of 001 had roughly a 67% chance of winning against 010.
This doesn't an 001 is the optimal solution, but as he calculated, in an eight-possible three-digit sequence (000, 010, 001, 100, 110, 111, 011, 101),
111 and 000 have the lowest winning probabilities. 011 and 100 both have an 87.5% winning chance.
anwhile, sequences like 100, 001, 011, and 110 counterbalance each other, forming a cycle of mutual checks and balances, reminiscent of a Rock-Paper-Scissors ga.
For a gambler, a 55% winning chance is already an imnse advantage. If you can push the odds to 67% or even 75%, it's almost reason enough to stake everything, with no reason not to follow through.
Claude said slowly, "Best of five rounds. Second round, your turn to choose first."
"..."
Hoenhem thought for a mont and slowly said, "I'll choose heads, tail, tail."
Although there's a cycle ford among sequences, 011 and 100 are relatively less constrained by probability.
Claude smiled, "Then I'll choose heads, heads, tail."
Indeed, Hoenhem's heart sank. The opponent's chosen sequence is probabilistically superior to his first-hand solution, but... it doesn't an there isn't a chance to turn the tables.
The results of the second round's tosses were.
tails-tails-head-tails-head-tails-head-head-head-head-tails-tails.
Another win.
Hoenhem's furrowed brows relaxed slightly; for his tails-head-head to win, the only solution was to throw three consecutive heads beyond the first three tosses to break the opponent's heads-heads-tail suppression and achieve his own result.
It requires double confirmation, and the winning probability isn't high.
"Wow, it really seems like luck is on your side."
Claude whistled, tilting his head with an eerie smile on his face, "It's already match point. In the third round, I'll choose heads-heads-head."
"Hmm?"
Hoenhem's eyes narrowed, peering deeply at the opponent. The odds of consecutive triple heads are low, and if he chooses tails-heads, the opponent's win rate could be drastically reduced,
In the first three tosses, you must directly toss "heads-heads-heads" to win—
Because once the sequence goes beyond the first three, it will inevitably have a "before heads-heads-heads, there is a tails" situation. No matter how you choose tails-heads-heads, you won't lose.
The win probability is...87.5%.
"I choose tails-heads-heads."
Under the cold probability calculation, Hoenhem has no reason to choose another answer, and gives the answer imdiately.
"Good, then I'll start."
Claude's thumbnail flicks the coin.
The round brass alloy disc rotates in the air, seemingly slow but quickly, eventually falling onto Claude's hand and being firmly caught.
The result shows... "heads"
"The first toss works pretty well for ."
Claude grins and ignores the dense surveillance spell formations in the void, placing the coin on his thumbnail again and flicking it.
Clang—
Again, "heads".
The atmosphere around Hoenhem suddenly changes. His 87.5% win rate is established on not tossing heads-heads-heads in the first rounds.
The opponent winning ans a probability of one-half multiplied by one-half multiplied by one-half, totaling only one-eighth possibility.
"Third round..."
Claude's fingers deftly flip the coin and toss it again.
The brass alloy coin spins actively and falls, with a snap being caught by his palm again.
"Heads."
"One-eighth probability, I win."
Claude casually whistles, "Fourth round..."
"Wait a mont."
Hoenhem raises his palm, "How about the next two rounds, I do the tossing?"
He's never been a superstitious person regarding probability. As a player, he also believes System wouldn't set an amiable old man with treasures in a high-difficulty dungeon as a considerate guide for players.
The opponent surely used so ans to trap him...
"Alright."
Unexpectedly, Claude nods generously and casually tosses the coin to Hoenhem.
As before, Hoenhem uses the invisible Mage's Hand to grasp the coin and scrutinizes it more closely with scout spell.
The portrait on the coin is severely worn, vaguely showing sharp facial contours, deep eye sockets, a prominent nose, and well-proportioned features with short curly hair,
From a general aesthetic view, he counts as a handso man,
vastly different from the three-ard creature Claude in front of him.
Hoenhem lifts his gaze and asks, "Whose portrait is on the coin?"
Due to the severe wear on the coin, he doesn't need to worry about [the character being highly renowned in the script world, causing Claude to suspect identity if unrecognized], as he can attribute it to being "indistinct".
"Who else? Tisiu, the pri among the seven pirate kings of the Endless Deep Sea, commander of the invincible fleet, symbol of all worldly wealth."
Claude seems sowhat puzzled, "I found this copper coin accidentally in this forest, presumably left when Tisiu and his subordinates ca to the Sacred Mountain for worship,
or perhaps brought in by so generation of travelers."
He looks at Hoenhem oddly, "You didn't think it's
on the coin's portrait, did you? Am I that ugly?"
"Uh, no, no."
Hoenhem suppresses the urge to retort, cognizant of the three-ard creature's extraordinary ugliness that would fit into horror ga scenarios in reality,
as a monster executing a jumpscare effect program, vastly different from the handso man on the coin.
The difference is probably like Duan Yanqing and Duan Yu...
But it might be normal too; in the ga of killing everything's possible, maybe Claude's race has a peculiar aesthetic.
Previously, there was a chubby player entering the script world's gloomy region, warmly welcod by a Dwarf Elf Clan who favored plumpness, taking out the skinny handso man in the team and leaving him, happily ever after—
Though the result was the player group falling into the Man-eating Spider's web, all being illusions, it did not affect the rule's effectiveness.
Don't easily discuss beauty or ugliness with natives from the script world.
Claude shrugs, "Last round I chose first, this round, your turn first."
Hoenhem takes a breath, "I choose heads-tails-tails."
re accident, before identifying the problem, the relation of superiority and inferiority in probability still stands.
Claude grins, "Then, I choose tails-tails-tails."
Again, like this!
Hoenhem's heart sinks, heads-tails-tails probability likewise superior to tails-tails-tails, maintaining the 87.5% win rate.
This round, he's the first chooser,
After choosing heads-tails-tails, the opponent can choose heads-heads-tails, achieving 67% win probability restraint, yet chooses tails-tails-tails with only an eighth winning chance.
Definitely, sothing went wrong sowhere...
"Since decided, let's begin tossing."
Claude glances at the floating coin, tilts his head, urging.
"...Alright."
Hoenhem narrows his eyes, the coin is tightly held by his Mage's Hand, the tossing power in his hands.
Mage's Hand is an extension of his own spell power, even more flexible and controllable than his palm.
'Let
see if you are cheating...'
The invisible hand flicks the coin, the coin spins on its own, cutting through a linear trajectory in the air.
Snap.
The entirely magic-composed Mage's Hand catches the brass alloy coin firmly at the sa tossing height.
Hoenhem's eyes flash with insight; he didn't cheat, but he already knows the result.
In the air, the surveillance magic spells infused with magic power focus intently on every rotation of the coin, calculating the result before it lands and is caught.
I've already won.
Hoenhem perceives the magic power leaking from his body (the low free energy environnt being extrely unfriendly to spell-focused players), unveils the coin at an incredibly slow pace.
Wait,
A snow mountain?
Hoenhem's eyes widen slightly, abruptly lifting his gaze to fix on Claude, voice low: "You're cheating."
"Did I?"
On the stone platform, Claude smiles: "You tossed the coin, and you caught it.
As a Mage, you're well aware if I interfered during the toss.
Don't spoil the situation too much just because you might lose..."
The three-ard creature lightly scratches his cheek with his right hand, leaving multiple blood marks, smiling: "Moreover, even if I did cheat, unless you have evidence to prove it,
the ga remains valid."
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