Exerpt from Dancing with Deception, or, My Travels Through the Third Circle, pg. 67.
…I emerged from the blackened maze to find myself at the very center of the two sisters’ realm. Twin thrones stood in repose, erected like sentries before two circular stairwells leading down. Perched upon them were the sisters, both alike. Their bottom halves were serpentine, long scaled bodies that coiled in bundles beneath them. From the waist up, they might have been human, but for their narrow faces and snake-like eyes. They were gaunt, and lay perfectly still as I approached, only their eyes moving as they followed me. They induced the distinct, electric feeling of predators waiting to lunge.
I stopped a goodly distance from them, and my guide flapped up to perch on my shoulder, whispering into my ear. “Master,” he said in his irritating, nasally voice, “these are the two sisters, for which this circle is named. You must pass their trial, if you insist on descending further into this madness.”
I simply nodded, and after a long pause, the imp continued. “One before you is Vajra, the other Majra. Behind them are two sets of stairs: one leads to the fourth circle, the other to perdition. Their trial is thus: you may ask a single yes or no question of one of them. Vajra will answer truthfully, while Majra will lie. With but this single answer, you must determine which stairwell to take.
“And which is which?” I asked. My companion, shrugged, his thin silver chains tinkling.
“None know.” he replied.
I pondered this trial, taking my time, as one should when dealing with the denizens of below. I could have forced my way through with seal and sword, but in the end I decided to play their game. There would be fewer metaphysical ripples to tell of my coming.
It was a simple game: a trifling, for one versed on the ways of fiends. I turned to the sister on the left. “Well met,” I said. “My question is thus: If I were to ask your sister whether the stairwell behind you leads to the fourth circle, would she answer ‘no’?”
The creature finally moved, winding back and forth in agitation. She spoke, the single word emerging as a low gutteral hiss: “Yesssss…”
I nodded, and proceeded toward the stairwell behind her, gripping my seal tightly lest they renege on their game…
“…huh,” Alexandria said. “That is similar. Why does that work?”
“Is that story true?” Rose cut in, talking over her. She wanted to grab the slim volume out of Robin’s hands and read the whole thing right then. It took an enormous effort of will to keep her hands inside her cloak, clutching at the cloth.
Alexandria turned to look at her. “Who knows?” she asked, seeming confused by the question. “It’s just a story.”
“King Solomon was a real historical figure,” Esper answered. “Whether he truly traveled through the nine circles of hell, who can say? Even if he did, I’m sure there are a thousand versions of that story. If the third circle he traveled through is the circle of lies–”
“It is,” Robin said.
“Then we already have two conflicting accounts. In the book there are two sisters, in the inscription there are three. In the book they speak Solomon’s tongue, in the inscription they speak their own.”
“Maybe Da and Ba are the words in Solomon’s language?” Rose said.
“No, Solomon spoke ancient Sumeric. This is known. Their words for yes and no were ‘heam’ and ‘inu’.”
“But which one is true?” Rose asked, looking back and forth. “Or is neither true?”
Alexandria shrugged helplessly. “Who knows?”
Esper shook her head. “Someone might, but not us.”
Robin tapped his chin. “I’ve read a lot about the circles. I’m not sure that question really makes sense.”
“What do you mean?” Rose asked.
Robin rolled his eyes. “I mean, I don’t think the circles are a real, literal place.”
“So they’re just a lie?”
“No, no. Just, like…argh, nevermind.” He slammed the book shut.
“No, wait! Just try to explain it once?”
Robin glared back at her.
“Fine, fine. Just let me think for a second.”
Rose realized she was clutching the cloth of her cloak a little too tightly. She released it, feeling her way through to look for her stone instead.
“Alright. You know math, right? They taught that wherever you grew up?”
“My mom did.”
“Mhmm. So you know division, then? Six divided by three is two?”
Rose resisted the urge to roll her eyes. “Yup.”
“Division is ‘real’, in some sense. It lets you accomplish things in the world. But it isn’t physical, it isn’t a place you can go.”
“But nobody disagrees on divison,” Rose said.
“If you read one book that said five divided by zero is infinity, and another that said five divided by zero is undefined, who’s right?”
“Well, uh…hm. I guess it depends how you want to do it. Saying it’s equal to infinity is maybe reasonable, but it causes some problems.”
“Exactly. Division is real, in some sense. Two different groups of people plopped down on opposite sides of the earth would both invent it. But they might define division by zero differently. Maybe one definition is better, but neither is lying.”
Rose thought about that, turning her stone over and over in her hands. “But the book was talking about a real, physical place. With a blackened maze and a seal and a sword and talking creatures.”
Robin shrugged. “I don’t know. Now can we drop it? I need to go put everything back in my trunk.”
Rose’s eyes darted to the book in her hands. “Can I read it?” she blurted out.
Robin raised an eyebrow.
“There might be other hints,” she said. “If it’s a whole book about the circle of lies.”
“No,” he said, flatly. “It’s delicate. Besides, I’ve read it five times. There’s nothing else useful in there.” He turned around and walked away. Rose resisted the urge to yell after him, try to convince him. It would just make him dig in his heels, probably.
Rose felt a hand on her shoulder. Alexandria gave her a friendly squeeze. “Let’s get back on track,” she said. “The two trials seem pretty similar. Why did Solomon’s question work?”
Rose shrugged, still distracted by the retreating book. Esper hummed brightly. “Borodin?” Alexandria asked.
There was no answer. Rose turned and saw he’d managed to sneak behind them and slump onto the ground again, snoozing peacefully. Alexandria squawked indignantly and marched over to wake him up.
The window would be best, Rose decided. Sunlight, fresh air, a view. It would be like when she’d snuck off into the forest to think, in her old life.
She needed to be alone to think. It had always been that way. Some people, like Alexandria, didn’t seem to be able to make progress without someone to bounce ideas off of. But Rose found that other people cluttered up her head and pushed out the finer thoughts.
She fished out her notebook from deep in her cloak. It was a bundle of papers wrapped in an oiled skin, about half full of scribblings, nestled in the thickest part of her cloak near the side where it would be safe from the rain. She took a fresh sheet and a thin piece of charcoal, inhaled a deep breath of the morning air, and got to work.
Why did King Solomon’s trick work? Precisely, specifically? She started stripping out details. He needed to ask a question that worked regardless of whether he asked Vajra or Majra. No, details: call them T and F, for truth and falsehood. That was the only thing about them that mattered.
He asked X, the first sister, whether Y, the other sister, would answer no to a question Q. Say Q had the correct answer A.
So there were four possibilities. X could be T or F, and A could be Yes or No. She checked all four possibilities, and yup, it worked. He always got the right answer.
Rose raised a hand to her face, rubbing her chin. That was an answer to why what he’d done had worked. There were four possibilities, and it worked in all four. But it wasn’t enlightening.
She fished through her cloak for her secret splinter of driftwood. It was different from her other treasures: the others were hard, cold, heavy, sturdy. The driftwood was light, airy. It would snap if she pressed too hard. She held it carefully, running her fingers over the tiny features.
The trick was Solomon had brought both sisters into the question, knowing that exactly one would lie. If you asked one sister what the other one would say, either the first sister would truthfully report the second sister’s lie, or the first sister would lie about the second sister’s truthful answer. Either way, the answer would get flipped exactly once.
That’s why Solomon had asked if the second sister would answer “No” – because the final answer would always be a lie.
She drew the question out on her paper. The true answer, A, was on the left, then an arrow took it through X, then an arrow took X’s answer through Y, and the final answer coming out on the right was always the opposite of A.
She was starting to feel like she understood the shape of things. She turned to the real problem.
The real problem was much harder, which was probably why they got three questions instead of one. The obvious thing to try was asking the first sister what the second sister would say if she was asked what the third sister would say – but because one of the sisters could choose how to behave, any chain of answers like that was unpredictable.
In some ways it was the opposite problem. The first one had been solved by constructing a chain of questioning that included both sisters, but the harder problem required a chain of questioning that excluded the one who could answer however she wanted. Otherwise the answer was always indeterminate.
She tried working it out by enumerating possibilities again. There were 3 sisters, so 6 possible orderings. The answer to the question was Yes or No, and the words Da or Ba could mean Yes and No or vice-versa, so that made…24 possibilities to check? She wasn’t even sure where to get started generating questions to check.
Maybe she could make the problem simpler? She didn’t even know how to solve the three-sister problem if they answered in normal speech, rather than their own language. “Work on the simplest problem you don’t know the answer to,” that was the advice her mother had given her. If the sisters answered in normal speech, there were only 12 possibilities…
Rose looked down at her paper, eye twitching. She wanted to strangle Robin. The notes in front of her were so brightly lit she had to squint, the sun almost directly overhead. She’d been working for hours, and it all could’ve been so simple, if only…
She checked it again. Solomon hadn’t needed to ask the first sister what the second sister would say. Doing that worked fine, but it was needlessly complicated. By putting both sisters in the question, he’d introduced exactly one lie. She’d been noodling around, trying to find a solution to the three-sister problem that didn’t involve asking questions that involved all three sisters, when she’d realized that you only needed one sister in the loop to use Solomon’s trick.
Just ask either sister “What would you say if I asked you whether the stairway behind you leads to the fourth circle?” If the sister he was talking to told the truth, then it was trivial to see that it worked. If the sister he was talking to lied, then the two lies would cancel out, and she’d end up answering truthfully.
It was a little twisty, but it worked. If you were talking to the sister that always lied, and you asked her “Does the stairway behind you lead to the fourth circle?”, and it did, she’d answer no. So if you asked her what she would say, she’d have to lie and say that she would’ve said yes. If the stairway didn’t lead there, she’d answer yes to the first question, so she’d have to answer no to the second.
Rose grabbed her hair. It translated perfectly to the three-sister problem, if they answered in normal speech. Just ask each of the three “What would you say if I asked you whether the leftmost passage leads to the tower?” The one that always told the truth and the one that always lied would both answer the same, and then if the third answer disagreed you could just throw it out.
Aaaaaaaargh. If she hadn’t heard that stupid story about stupid Solomon and his stupid clever tricks, she wouldn’t have gotten so caught up in trying to design cycles of asking sisters what other sisters would answer if they were asked…
Rose took a deep breath. She released her hair, noticing she was also squeezing her driftwood a little tightly with the other hand. She should take a break. This was good progress.
She tucked her driftwood away and stood up, stretching. It was almost lunchtime anyway. After a moment’s thought, she tucked her notebook away as well. It never hurt to be safe.
Her bag was still in the corner where she’d slept. She fished out some pemmican, grabbing extra for Robin, and took it to the fountain to eat. Robin was already there, sitting alone with Alexandria. They were each holding an end of stale-looking bread, speaking quietly.
Their conversation trailed off as Rose approached. Robin glanced up at her, then looked away. Rose felt a little clench of apprehension, but Alexandria looked up as well with a beaming smile. “Any progress?” she asked.
“A little,” Rose replied. She sat down next to Alexandria, on a diferent side than Robin, and leaned over to take a sip of the water. She was thirstier than she’d realized.
“You’re going to burn, sitting in the window like that,” Alexandria commented.
Rose looked down at her hands. The backs were already a little pink, and her nose probably looked similar. “Oops,” she said.
Alexandria giggled, holding a hand to her mouth. “How do you stay so pale living in the country, anyway?” she asked.
“It rains a lot around here,” Rose said. She looked past Alexandria at Robin, who was still avoiding her gaze. “And the forest is old.”
“The forest is old?”
“The trees are tall and wide, they block out a lot of the sun. Some of it is coppiced near town, but where my Dad and I live it’s old and dark.”
“You don’t live in town?”
“No. A few miles out.”
“My dad’s a woodworker. Does odd jobs sometimes. My mom…” Rose trailed off, looking down at her hands. Her fingers were getting greasy from holding the pemmican.
Alexandria smoothly changed the subject. “Dexter’s finally up. He slept almost until noon.”
Rose looked up. A giggle escaped her lips. “Who sleeps that long?”
Alexandria broke eye contact, looking over Rose’s shoulder. “I don’t know. Not even Borodin, I’ll tell you that much.”
“Maybe he was up late?”
“I was,” said a voice behind her. Rose jumped a little. She turned to see Dexter emerging from the rest area. He looked surly, but his eyes were sharp. “Insomnia.”
“Oh,” Rose replied. “I’m sorry.”
Dexter waved his hand. “I’ve never been able to sleep right,” he said. “My thoughts tend to race, and they don’t like slowing down.”
“Same! Oh my gosh, I know exactly what it’s like. I sleep fine, though. I just wake up in the morning with my head already whirling.”
Dexter smiled with half his mouth, his expression hard to parse. “Lucky you,” he said. “Figured out where we’re going for us yet?”
“She’s close,” Alexandria broke in. Rose felt her cheeks heat up. That wasn’t quite what she’d said.
“That wasn’t quite what I said.”
“How close are you, then?”
“I think I know what I’d ask if they answered in our language.”
“And what’s that?”
Dexter leaned over and splashed some water from the fountain onto his face. Great, he could use it.
“Er, you ask all three of them, ‘What would you say if I asked you whether the leftmost passage leads to the tower?’, and take the majority answer.”
Dexter paused for a moment, water streaming through his hands.
“And that works?”
“Yup. I checked it. There’s twelve possible cases, and…”
“Aren’t there only six?”
“Yeah, 6 configurations of sisters, but then there’s 2 configurations of paths.”
“Right.” Dexter straightened, flicking water off his hands and blinking it from his eyes. He didn’t look even a tiny bit cleaner. “Let me see.”
“You can tell it works without checking them all because–”
“No, let me think. OK. Yeah, it works. That’s not a yes-or-no question though.”
“‘What would you say’ isn’t a yes-or-no question.”
“It is when asking what you’d say to a yes-or-no question.”
“Maybe. But one sister can answer freely, right? She could just say whatever. And it seems a touch fragile, even in the other case.”
Rose opened her mouth to argue, but stopped herself. Precision, right? That was the key to these things? It was pedantic, but he was right. In the case where one of the sisters could answer whatever she wanted, asking what she would answer wasn’t a yes-or-no question.
“Hm. Then…I guess the question is, ‘If I were to ask you whether the leftmost passage leads to the tower, would you answer yes?’.”
“Mhmm.” Dexter said. His eyes were fixed off in the distance, above Rose’s head. She turned to look, but there was nothing there. When she turned back to him, he was moving his fingers awkwardly, curling and uncurling them like he was counting in some strange system.
“Well,” he said after a moment. “That was easy, I guess. Good job.”
“That’s the whole trick. You can’t ask whether they say yes, because they don’t answer in our language. So you ask ‘If I were to ask you whether the leftmost passage leads to the tower, would you answer Da?’ and you’re done. If a majority say Da, you take the left passage.”
Rose opened her mouth, thought for a moment, then closed it. It couldn’t be that simple, could it? But it had the same sort of double-negative-stacking shape as the original “what would you say if I asked you” trick – if Da meant Yes, it was the same question she’d asked, and if Da meant No, then the answer to the nested question would be No but so would the answer to the outer question…
She shook her head, the line of thought was a little bit too tangly to hold on to. “Are you sure it works?” she asked, looking around. She caught sight of Alexandria next to her, remembering the other girl was there. Alexandria had her chin propped up on her fist, brow wrinkled. Rose recognized the confusion, reaching instinctively for her shard of glass to help explain it to her…but wait. That was weird. Why wasn’t she already holding it while she talked to Dexter?
“Of course it works,” Dexter said. She turned back to him. He had his hands in his hair, trying to untangle a wet knot, a scowl on his face.
“How do you know?”
“It’s only twenty four cases. I just checked them.”
“Spectacular!” Borodin said, clapping Rose between the shoulder blades.. She had to stumble forward a step to keep from falling.
“Borodin,” Alexandria chided.
“Sorry, sorry. She’s a tiny thing. But as I was saying, excellent work!”
Rose felt her cheeks warming for the second time that day.
“She blushes a lot, too,” Borodin commented. Alexandria gave him a sharp jab in the ribs, which he barely seemed to notice.
“Er…” Rose said, standing up and straightening her cloak. “It was, um…Dexter was the one who solved it.”
“That lout?” Borodin asked, agape. He seemed personally affronted by the idea.
Alexandria elbowed him again. “Be nice.”
Rose sighed. Borodin seemed to have built up energy from sleeping through the morning. He began pacing back and forth, stroking his beard, a kind of bounding energy in every step.
“I, uh,” Rose said, tracking him back and forth with her eyes. “I double-checked it. Alexandria did, too.”
It had taken the other girl entirely too long, in Rose’s opinion. They were mere hours from twilight now. But she supposed she couldn’t begrudge a little caution.
“That’s good enough for me,” Borodin said. “Who wants to do the honors?”
Everyone’s gaze turned to Alexandria, except for Esper, who turned her head very intently toward an empty patch of wall. The blonde girl brushed her hair back, suppressing a smile. “I’d be happy to,” she said, turning toward the faces on the wall.
Rose took a deep breath, reaching for her riverstone. If they’d gotten it wrong, they’d have to guess, and fully half the time they might…die? It didn’t feel real. She suddenly wasn’t so annoyed that Alexandria had taken her time making sure of their answer.
Alexandria walked up to the face furthest to the left, leaning close to its carved ear, and spoke to it without a hint of fear on her face.