Everyone else seemed to share the same thought.
“Ren,” Lin called, breaking the tension, and Ren quickly responded, still dazed from the previous exchange.
“Y-yeah?”
“What’s your estimate for the time it would take for the beast to reach the point of ‘entry’ and for us to reach the same point via the border?” Lin asked, strolling to the edge of the [Snowy region], his hand still clutching his stomach.
He stopped directly at the border between the [Snowy Region] and the [Swamp region], then turned, gesturing across the border to clarify the exact paths he was referring to in his question:
- From the center of the [Snowy region] to the border of the [Swamp region], the “shortest path.”
- From their current location to the point where the beast would potentially enter the [Swamp region].
Ren glanced at his screen for a moment, a thought suddenly striking him.
“Since it’s kind of a right-angled triangle, we could use Pythagoras,” he muttered, prompting a few among the crowd to grimace at the sudden mention of a math term. Oblivious to their sourness, Ren’s fingers traced across his status window, drawing a few lines to form a triangle. The status window updated in real time, reflecting his calculations as though it were a digital chalkboard.
That’s useful. The group collectively marveled at the windows functionality, imagining the nightmare it would’ve been to explain the concept through words alone.
“But we only have the hypotenuse… which is about 10 minutes minimum. How do we…” Ren suddenly trailed off, jotting down a few variables on the screen as he fell deep into thought.
He’s panicking. Lin observed silently, attributing Ren’s scattered thoughts to the pressure of the situation.
They had previously estimated that the time the beast would take to travel from the center of the [Snowy region] to their current location was around 10 minutes, representing the hypotenuse of the triangle (path c). However, without another known value, applying the Pythagorean theorem was impossible – they needed at least one more variable to do that.
“Mmh, maybe we can use some trig functions? But we’d need the angle…” Ren muttered after a moment, mentally shifting gears and expanding on his drawing.
If the paths formed a right-angled triangle, they could apply trigonometric functions even with just one side’s value. However, another problem arose: determining the angle between path c and path a—that is, the longest path from the center of the [Snowy Region] to their current location and the path from their current location to the possible point of entry at the border of the [Swamp Region].
Thinks too far ahead and makes simple mistakes, Lin thought as he watched Ren.
The others seemed impressed, especially those who could follow Ren’s reasoning. Although they vaguely remembered such concepts from their high school days, they never imagined they’d be useful in a situation like this.
Noticing Lin’s disappointed expression, another figure stepped forward—a golden-haired boy dressed in a school uniform.
Stolen from its original source, this story is not meant to be on Amazon; report any sightings.
“If the area is as symmetrical as the map suggests, then…” Isadore said, erasing some of the details from Ren’s screen and drawing two simple lines.
“We have an isosceles right-angled triangle,” he explained simply.
The realization hit Ren a second later as more lines appeared on the screen, almost as if mirroring his thoughts before he could even write anything.
“R-right, so the two distances would be the same, meaning we only have one unknown variable,” Ren said, slightly thrown off by Isadore's quick thinking. If their maps accurately reflected the area, and given that all regions were the same size, they could deduce that any path from the center of the region to the midpoint of any side of the region would be equidistant.
This symmetry made it easier to apply the Pythagorean theorem.
The distance from the center of the [Snowy Region] to the border of the [Swamp Region] through the shortest path would be equal to the distance they would also need to travel to reach that point of entry from their current location.
With only one variable left to calculate, reaching a conclusion became straightforward.
“Should be around seven minutes, give or take,” Isadore muttered, bringing a smile to Ren’s face as he nodded, having reached a similar conclusion.
“Mmh, you’re really good at this,” Ren said, praising Isadore, who gave a small chuckle.
“Well, I’m still a student, you see,” Isadore replied, trying to downplay the compliment.
However, everyone could immediately tell he was just being humble. While being a student might mean he remembered most of the theories off the top of his head, not every high school student would have been able to think of applying them at a glance—especially under such intense pressure.
“Correct,” Lin interrupted, cutting short their bonding moment. Though he didn’t even have a status window open, everyone could somehow guess that he probably already knew everything they were discussing. Sensing their gazes fall on him, he continued,
“It might be seven minutes at best, but we should assume it could take the beast even less time, especially if its primary objective is to move while keeping the five-minute forced teleportation limit in mind,” he explained. Both Ren and Isadore nodded in agreement.
“It’s definitely possible if it disregards its own body entirely,” Roben interjected from the sidelines, still pinning the blonde woman, who wore a bored expression, to the ground. If the beast abandoned all rationality and aimed solely to reach them, it would be unwise to put too much faith in their calculations of the estimated times.
“Yeah, the chance is very high,” Ren muttered, his focus on his status window.
“That’s only surface-level thinking,” Lin muttered, drawing their attention. He walked up to Ren, standing near his status screen as he added,
“If it were me, I wouldn’t go there directly.”
“Mmh?” Ren mumbled, a spark of curiosity in his gaze.
“Heading toward the midpoint would be the best choice,” Lin said, tracing a new path on the screen—from the center of the [Snowy Region] to the midpoint between their current location and the point where the Minotaur was presumed to be heading.
“But how long would that take?” Ren asked, noticing that while it cut through both the longest and shortest paths, it still appeared somewhat lengthy.
“Since it’s the midpoint, we just divide the value we found—seven—by two to find the shortest path, which would be 3.5. Using that and the previous hypotenuse of ten, we can find the other path using the cosine rule,” Isadore muttered, already working on it as he began drawing on Ren’s status window.
“What about the angle?” Ren asked, as if trying to recall the theory Isadore referred to.
“It’s 45,” Isadore replied, pointing to the triangle. “It’s an isosceles triangle, so we already know the base angles are equal. And for a right-angled triangle, since one angle is 90, the other two would be 45 each using the equal sides, equal angles rule,” he explained simply, his fingers gliding across the screen to perform the calculations for everyone to see.
Since Pythagoras’ theorem only applied to right-angled triangles, they needed an alternative approach for other types of triangles. In this case, knowing the lengths of two sides and the angle between them made the law of cosines the most suitable method. Given that 45 degrees is a special angle, the calculations were straightforward even without a calculator.
“Eight minutes tops,” Isadore concluded, his tone calm as everyone watched in awe. Oblivious to their gazes, he continued,
“The deviation caused by aiming for the midpoint adds just one minute, but we could probably target the midpoint of the two midpoints to reduce the deviation further while still keeping the other path within the five-minute range. I estimate it would take about seven minutes and 30 seconds to reach the swamp region, and at least five minutes from there to reach us,” he said, starting to mumble.
His increasingly technical words began to lose some of them, and they just nodded, trying to follow along despite the details slipping past their comprehension.
He’s quick on the uptake, Lin thought as he watched Isadore with a small smile, following his train of thought.