Chapter 28
This was falling apart. And quickly.
The poor woman, stressed about her father (for some reason) and the TikTok video, was now trying to deal with wrapping her mind around the idea of ten dead bodies.
Vaughn had seen it in Ivy’s face.
Aaron Treadman was likely her first.
The first corpse that Vaughn had seen had been his grandmother’s—Betty Ryan. His parents had talked about an open casket, but he, being only seven at the time, had no idea what that meant.
It had been a rude awakening, seeing the woman who lying in a coffin so still—impossibly still.
Ivy Reeves was no seven-year-old, but that didn’t make it much easier. Darnell . . . God damn it, it had been his idea to come here.
“Dr. Reeves—I mean, Ivy—we shouldn’t have bothered you.”
“Ten boxes, you said?”
Ivy was deep in thought and her brow knitted.
Darnell: “Yeah.”
Vaughn just wanted to leave the woman alone, but Darnell was blocking him in.
“Numbers on their chests, numbers on the boxes, different numbers in the boxes?” Ivy said. Her voice was different somehow. Far-off.
“Well, it’s the same ten numbers repeated. Uh, two, three, seven . . . thirteen.”
“Two, three, five, seven, eleven, thirteen, seventeen, nineteen, twenty-three, and twenty-nine.”
“How’d you know?” Darnell asked.
“Those are the first ten prime numbers.”
“Ah.”
“The boxes . . . ?” Darnell pressed. “Any idea what this all means?”
Ivy didn’t answer at first. Her brow still stitched, she ran a hand through her hair. It got stuck halfway and she teased her fingers free.
“Ten numbers, ten boxes, ten . . .” Ivy trailed off before saying the word “victims.”
“Yes,” Darnell confirmed. He kept shooting these looks at Vaughn, which he found particularly annoying.
“Have either of you ever heard of the 100 prisoners problem?”
Now it was Vaughn’s turn to stare at Darnell. The big man shrugged.
“Can’t say that I have,” Vaughn admitted.
Ivy’s forehead softened, and the ‘11’ fold between her eyes disappeared completely. When she spoke again, she seemed more or less steady.
It was as if her brain switching into math mode had made her forget all about Aaron Treadman.
“It’s a classic probability theory in the form of a game.
One hundred prisoners are placed in a room and are given the rules: each one is assigned a unique number from one to one hundred, consecutive.
They are all to enter an adjacent room alone, one at a time.
They can discuss strategy beforehand, but once they are in the room, they can no longer communicate with the other players.
The room has one hundred boxes in it, all labeled consecutively, again, from one to one hundred.
They can open up to fifty boxes, and that’s it.
The numbers inside the boxes have been randomized.
Their goal is to find their number—the one assigned to them—in one of those fifty boxes they open.
If they do, they have effectively ‘won.’ The boxes are closed again, and the next person enters the room.
Same rules apply. If all one hundred find their number, they’re released from prison.
If even one of them fails, they all lose. ”
“They die?” Vaughn blurted.
“Yes. These are prisoners, remember? Anyway, it’s just a hypothetical game. Win the game, you live. Lose, you die.”
“I mean, depending on the sentence, it seems like a good deal. Fifty-fifty, right?” Darnell said, playing along.
“Not exactly,” Ivy continued. Vaughn recognized her now as the same woman in the video.
Confident and sure of herself, despite the uncomfortable analogy she’d used on TikTok.
“For the first person to enter the room, the odds are, like you said, fifty-fifty. But for every one of the one hundred prisoners to find their number? Opening only half the boxes? The formula is one in two to the exponent one hundred.”
One of Vaughn’s eyebrows lifted. He felt impossibly stupid at this moment.
“It’s . . . not fifty-fifty?”
Darnell sounded impossibly stupid at this moment.
“No. It’s more like less than one in a decillion.”
What the fuck is a decillion?
Vaughn had just recalled decuple, as in decuple homicide. He wasn’t in the mood to learn another number, and decillion sounded somehow even more ominous than decuple.
“A what?” Darnell asked.
If Darnell Sacker had a superpower, it was this: the ability to not give a shit how he looked in front of others. He just wanted to understand, to find the bad guy. He would ask question after question until he exhausted a suspect.
He wouldn’t break down, wouldn’t crack.
Never.
Vaughn had seen Darnell run a sixteen-hour interrogation by himself without so much as a break for a drink of water. Went against every rule the PPD had, but their suspect had eventually confessed.
“It’s effectively zero,” Ivy said. “The probability of all one hundred prisoners finding their number is zero.”
“Well, I take back my previous answer. I’ll take my chances behind bars.”
Darnell’s secondary power was humor. And his damn hunches—Vaughn couldn’t forget those, either.
“But there weren’t one hundred boxes or one hundred prisoners. There were only ten,” Vaughn said, trying to get them back on track. “And they weren’t prisoners.”
Until they were.
The digital locks on the doors in the barn flashed in his mind.
“The odds improve with fewer participants.” Vaughn noted that Ivy wasn’t viewing them as victims anymore.
Ivy had turned this into a true math problem, her way of dissociating herself from the image she’d seen on his phone.
“But the probability of success is still only 0.1 percent with ten prisoners.”
“What’s with the—”
Vaughn wanted to say prime numbers, but Ivy wasn’t done yet.
“But that’s only if each contestant opens random boxes. If all of the prisoners utilized a permutation approach, they can vastly improve their odds of success.”
“Permutation approach?” Darnell asked.
“Yeah, it’s simple, really. Say you’re prisoner number three.
The first box you should open is the third one.
Find your number by sheer chance? Great.
You’re done. If you don’t—instead, you find number seven inside, for instance—then you go to the seventh box.
Open that box. Repeat.” Ivy drew a small circle in the air with her finger.
“You form these permutations or loops. Using this strategy can vastly improve your odds. In the ten prisoner scenario, the odds of being successful goes from 0.1 percent to about 36 percent.”
“36 percent?” Darnell was legitimately surprised, but him pretending to understand was a farce.
Ivy nodded.
“Yep. About a third of the time, you can beat the game.”
Darnell whistled, laying it on thick.
“But the numbers aren’t one to ten,” Vaughn remarked. “They’re prime numbers.”
“Doesn’t matter what the numbers are. Could be random, could be prime—as long as they’re all the same, you can create loops. I’m guessing that’s why the numbers were on the outside of the box, so that the contestants could follow the permutation approach.”
Vaughn thought about Aaron Treadman lying on the gurney at the morgue, Dr. Button hovering over him.
Aaron Treadman had a high school education. He’d been a security guard at Princeton, but was unemployed at the time of his death.
Vaughn put the odds of Aaron knowing the “permutation approach” for solving the 100 prisoners problem at pretty close to zero.
One in a . . . decillion, maybe.
Hey, maybe he wasn’t beyond learning new things.
Darnell shrugged.
“I don’t . . . really get it, but okay.”
Ivy blushed.
“Sorry, I nerd out sometimes. Math—”
“No, it’s okay,” Vaughn cut in. “But why prime numbers? Why not just use one to ten? Would make things easier, wouldn’t it? Like to organize the game?”
Now Ivy shrugged.
“I have no idea. But, yeah, it would make things much easier from a design perspective.”
“Hmm.” Vaughn took all this in, realized that both Darnell and Ivy were waiting for him to say something. “Well, Ivy, thank you for your help. Again, I’m sorry about the photo.”
“No problem.”
With a curt nod, Darnell and Vaughn left the professor’s office.
“You understand any of that?” Darnell said out of the corner of his mouth as they walked the hall.
“Not a fucking word.”