Touhou Project by Shanghai Team Alice
Knights and knaves accredited to Raymond Smullyan
In the sealed, secret world of Gensokyo, a shocking change has overtaken the land and its people. Through the application of a forbidden template, all of the inhabitants have now found themselves either unable to lie, or unable to tell the truth.
All of this was the result of one Yukari Yakumo, the Youkai Sage, reading a certain book of logical exploits by one Raymond
Now, an inhabitant who finds him- or herself unable to lie is called a knight, and an inhabitant who finds him- or herself unable to tell the truth a knave.
Remilia Scarlet, the Scarlet Devil, has awoken this night to find herself unable to lie, and therefore a knight. So not only is she Queen of the Night, she is also Queen of the Knights. She has also discovered that her faithful servant Sakuya Izayoi, the Dangerous Illusionist, is a knave, who always lie.
It seems that it is up to Remilia to solve this incident, to return lies and truth to equal distribution amongst Gensokyo's inhabitants.
After breakfast, of course.
Knights and Knaves:
A Logical Adventure of Gensokyo
by Wyrm
Chapter 2:
In Which Remilia Scarlet Questions her Household
The Scarlet Devil Mansion
"What is all this litter?" Remilia Scarlet asked, kicking at scattered beige pamphlets over the otherwise pristine crimson floor. No sooner had she stepped out of her room wearing her pink dress and pouffy hat, she had discovered this mess littering her precious mansion. Her black wings twitched with annoyance at the sight.
"This mess was here when I entered with your breakfast," commented Sakuya, which was, of course, a lie.
"So it had appeared suddenly," Remilia remarked as knelt down to scoop up one of the pamphlets. It was almost ordinary, with two folds dividing it into equal thirds. It had a title proclaiming, "Knights and Knaves: What They Mean To You!" Opening it, Remilia saw that the pamphlet explained, in condensed form, what both the book and the letter explained. Anyone reading the pamphlet would get the basics of the current incident, and a few test questions and statements would reveal what type you were.
"No doubt this is the Gap Hag's doing as well," Remilia (a knight) remarked aloud.
"We will have to explain what 'knight' and 'knave' mean to everyone we meet," stated Sakuya (a knave) falsely.
It still took some effort for Remilia to ignore Sakuya's lies and see the truth behind the lies, but she was getting there. "How convenient!" she said, giving the impression that she never even noticed Sakuya was lying. "And, as it happens, we have found our first group of maids to ask."
1. The First Dance of Fairies.
The fairies were mumbling over one of the small pamphlets that were now scattered throughout the mansion, and probably the whole of Gensokyo. By the end of the day, everyone no doubt will have a basic grasp of the situation. Probably even the illiterate would get the idea from explanations of their peers.
Remilia crossed her arms and brusquely addressed the three maid-fairies. "You three fairies. I want some information from you." The three cut all their conversation short and obediently stood to attention in line, as was the habit of the mansion.
Remilia stepped up to the first one. She was, of course, one of countless multitudes of fairy maids in the mansion, but even if Remilia did know her personally, there was a new feature to her that she needed to ascertain.
"Are you a knight or a knave?" she asked this first fairy.
The fairy answered, but mumbled indistinctly. Remilia didn't want to get into a one of those horrid cycles of badgering a shy maid to speak up, which would lead her to answer even more quietly. So she turned to the second, asking, "What did she say?"
The second fairy chirped, loud and clear, "She said that she was a knave."
The third shouted, "Don't believe her! She's lying!"
Remilia realized pretty quickly that her question to the first fairy would not have revealed anything probative even if she had heard it, being that knights and knaves alike would claim to be knights. She was, however, able to tell the types of the other two. It was very similar to Smullyan's first knight/knave puzzle, after all.
Solution: No inhabitant of Gensokyo can now claim to be a knave. For a knight could not falsely claim to be a knave, and a knave could not truthfully claim to be a knave. Therefore, the second fairy is a knave by claiming that the first fairy said such a thing. The third fairy is contradicting her, and therefore is telling the truth and a knight.
2. What Type is the First Fairy Maid?
After determining the other two fairies' types (that is, whether they were knights or knaves), Remilia still needed the type of this first fairy. So she addressed the first again. "You, answer this and speak up this time. Are you a lying knight?"
"No," came the first fairy's clear answer.
Remilia was immediately able to tell her type. She also knew such a trick would not work with anyone but her own staff, but Remilia supposed she would pick the low-hanging fruit when she could.
Solution: There is no such thing as a lying knight, so a knight could not honestly claim to be one. But a knave would, since she isn't any kind of knight, much less a lying one.
The trick would not work with just anyone, because Remilia knows that there is no such thing as a lying knight. As per (presumedly) Ran's letter stated, someone not of her staff, and therefore had no prior compulsion to obey her orders, need not answer a question Remilia already knows the correct answer to!
3. Dismissing the Knave Maid.
Remilia now knew she had two fairy knights and who they were, and as such were reliable sources of information: whatever information they gave would be considered as good as the truth. The fairy maid who was a knave was no longer of any use.
Remilia turned to the fairy maid who was a knave and told her, "You are dismissed."
The fairy hesitated. She waffled around, looking distraught and unable to decide what to do.
Remilia frowned at her, and spoke again, this time sharply. "Didn't you hear me? I said 'you are dismissed!'"
The fairy jumped, but quickly perked up and smiled. She happily moved to clean up the mess that the hallway had become.
Remilia was puzzled for just a moment why the maid was happy and obeyed so promptly the second time, but smacked her head as the answer came to her. Of course she wouldn't know what to do the first time!
Solution: Remilia had not yet proven she was a knight! 'You are dismissed,' is a statement, and if Remilia were a knave, would imply that the fairy knave was NOT dismissed. But Remilia had said the statement, and the fairy heard perfectly well, so when Remilia stated that she had said the statement, word for word, the fairy knew it was true and therefore her mistress was a knight.
—/—
The Great Unmoving Library
"This is going to be tricky..." Remilia mumbled, nipping at her finger.
"I imagine Lady Patchouli would be of no help to us," Sakuya lied.
Remilia was already getting used to the flopped nature of Sakuya's advice. However, the two maids gasped in shock at Sakuya's statement. "The head maid of the mansion is a knave," Remilia found herself volunteering.
The fairy maids perked up immediately at that news, and tittered at the discovery.
Sakuya acquired a dangerous look. Producing one of her knives, she addressed the two maid knights, "You will not stop that, and I will not put one of these into an uncomfortable place." The two fairies deciphered the logic quickly and sobered up immediately.
"Enough of this banter! Let's get down to business!" Remilia cut off any further 'discussion' or threats. "Does either of you know if Patchouli is a knight or a knave?"
The two glanced wordlessly at each other, then back at their mistress and boss. "We don't know if Patchouli-sama is a knight or a knave," the first fairy knight said.
The second fairy knight added helpfully, "But we do know that, of Patchouli-sama and Koakuma-san, at least one of them is a knight."
Remilia remembered the advice of the letter, that if the two maids were unsure of the truth of a statement, they could not make the statement. Therefore she could count on at least one of Koakuma and Patchouli being a knight. "Maybe that will be enough. To the library, Sakuya," announced Remilia, dragging Sakuya behind her.
The fairies Sunny Milk and Luna Child sighed in relief as the two left their sight. "Good thing they didn't ask more probative questions, or we might have been in deep trouble. It's hard to pull pranks when you can't lie your way out of tough spots," said the trio's leader. They moved to pick up Star Sapphire. It was obviously time to leave.
4. Is Patchouli a Knight?
The doors to Patchouli's great library, Voile, opened with a loud creek. "Sakuya, get someone to see to oiling this door's hinges," she remarked.
"It will not be done, m'lady."
Remilia sighed for the umpteenth time.
"Patchy! Patchy, where are you?" Remilia called out. She immediately thought better of it, "Never mind! Don't answer that! Just call out to us!"
Remilia strained her ears and heard a faint call from the back of the library. It told her everything she needed to know — Patchouli's location. The beauty of it is that she didn't need to know if Patchouli was a knight or knave.
Shortly thereafter, she found Patchouli sitting at a reading table surrounded by books. Koakuma arrived much the same time, carrying a huge stack of books which she just placed on the table. "Ah, there you are, Patchy!" declared Remilia, calling attention to her and Sakuya.
"Remilia," murmured Patchouli.
"So sentence fragments don't seem affected either," Remilia mused. Patchouli moved her head minutely. Remilia couldn't tell, but it looked like a nod. Damn. "You two, make some statements about your type," Remilia said.
Koakuma immediately volunteered. "I am a knight."
Patchouli huffed, removing her glasses. "If that were true, then I am a knave," she stated flatly.
Remilia was able to tell the types of Koakuma and Patchouli with all the information and statements provided thus far.
Solution: We had been told (by knights) that at least one of Koakuma and Patchouli is a knight. If Koakuma is a knight, then her statement is true. Then if Patchouli's statement were true, it would imply that she were a knave and therefore a knave who has made a true statement. Alternately, if Patchouli's statement were false, then the only way it could be false if she were not a knave, therefore a knight and therefore a knight who has made a false statement. Either leads to a contradiction, and therefore Koakuma is a knave. Since there is at least one of the two is a knight, Patchouli is therefore a knight.
5. The Maids Were Useless!
A moment later, Remilia shouted, "ARGH! Those fairy maids' information was bloody useless! I could have deduced your types just fine without it!"
Solution: An implication is only false if the antecedent (the 'if' part) is true but the consequent (the 'then' part) is false. If Patchouli were a knave, then the consequent is true, and therefore her entire statement is true, and we have a knave who made a true statement. Therefore, it is not possible for Patchouli to be a knave; she must be a knight, and her statement is true. The only way for a true implication can have a false consequent is if the antecedent is also false, and therefore Koakuma was lying and thus a knave.
—/—
Discussion Within the Library
"I don't think we can count on danmaku being the solution to this problem," said one Patchouli Knowledge, the Unmoving Great Library of the mansion and a knight. "If the culprit is Yukari as you suppose (and I don't think it's unlikely), it would be a challenge only soluble by using logic, rather than one using bullets. Otherwise, there would be no point sorting us into knights and knaves. Presumably ourselves as knights and knaves can be messed up exactly as easily as our normal selves."
"I agree," Remilia replied. "This isn't to say that danmaku will not be useful, however. One can still imagine troublemakers making things difficult."
Patchouli and Remilia sat across from each other while the two knaves commiserated. Remilia had lost interest in following the conversation, as it consisted completely of lies. It turned out that the books Patchouli had surrounded herself with were ones on logic and logical puzzles instead of her normal tomes on magic.
"Of course." Patchouli abruptly segued from a previous subject: "As you surmised, it seems nothing but a complete statement is affected by type. After all, only statements can be said to be true or false. A command really isn't true, nor is a question, even a rhetorical one. Sentence fragments aren't either, for their meaning is incomplete. Some exclamations can't be said to be true either, though I'm not too sure about them because, given some manipulation, they can be reformed into proper statements, and so fall under the type restrictions."
"So questions and commands can be used without fear?"
"Almost," Patchouli said, leaning back in her chair. "Some commands take the forms of statements, as you discovered from the fairy knave. Further, some questions are really statements in disguise. We'd still best be careful.
"You need to study up on your copy of Smullyan's book, as well," continued the Girl of Knowledge and Shadow, "The puzzles have so far been simple. No doubt they will become more mind-boggling and frustrating for you, as the puzzles in the book had."
Patchouli took a moment to check her notes. She had noted that her handwriting seemed unaffected, but she had been unable to write lies with her right hand. She had assumed this was due to the fact that she was a knight, but Ran's letter made her not quite sure of that anymore. "Now, you said the letter said that you needed to find the types of every household member, neglecting the fairy maids — and of course you're a knight. Don't be so quick in revealing that, by the way. Some of Smullyan's puzzles require you to keep your type a secret, and some challenges may require it as well." The youkai magician held up her hand, fingers but not thumb spread, and ticked them off one by one. "You know the type of yourself, Sakuya, myself, and Koakuma. That's four of the people you need to type. Two remain."
"I'll leave Meiling to last. She's so dumb she'd probably reveal herself as soon as she opens her mouth." Remilia's face was grim at the task looming before her. "That leaves... Flandre. Right now, she's either telling lies to herself, or she's telling herself the truth. Either possibility is frightening."
"She'll probably be in no mood to play logic puzzles, either," observed the witch.
"I might get killed before I even find out if she's a knight or a knave."
The silence passed for a few minutes, as Patchouli thought about the matter. "Perhaps the fairy maids have some information," she said. "They may have heard her mumbling and thus be able to deduce her type." She moved to her great stack of books and started researching, stating in passing, "This may take some time, and you have a book to read."
Remilia collected the small, orange, ragged paperback and stood. "Fair enough. I'll return later."
—/—
Fairy Twins
Remilia spent much of the evening sharpening her brain on the strop of Smullyan's book. She wrestled with the various logical concepts Smullyan presented in the guise of logical puzzles, up to and including the subtleties of Gödel's theorems. If she wanted to best Yukari's challenges (if they were Yukari's challenges), she needed to know this stuff cold.
She did not finish the book in one sitting, but Remilia managed to plow through better than a third of it before her mind needed rest. Perhaps now Patchouli had some more information on how to determine Flandre's type.
"Patchy!" she called as she entered the library, "What have you found out?" Hearing no answer, she walked to the reading table where she last saw Patchouli sitting. The magician wasn't visible, but there was a large pile of paper in a disorganized mountain of writing and wood pulp covering Patchouli's chair.
Remilia stared at the mountain of manuscript for a moment. That looked like it could contain...
"Mukyu!" the mountain squeaked, and shuddered. A few leaves of paper shook off and began fluttering down around it.
"Patchy?" Remilia yelped, leaping onto the pile and scattering papers off of it to free her librarian. Slowly, the Locked Girl emerged from the mountain. "Please be careful, Patchy. I don't want you to smother yourself."
"Or concuss myself," added the purple-haired youkai magician, "Darn thing got too tall and fell on me."
"So what did you find out?"
"The first thing I found out that is that Koakuma can write truths —and only truths— if she uses her left hand, while the other can write only lies. Same with Sakuya," Patchouli began.
"Patchy! You're supposed to be finding out stuff about my sister, not wasting time with what a knave can or cannot write!" raged Remilia.
"I'm getting to that. Being able to write truths meant that I could use them to gather statements from our fairy maids," Patchouli explained.
Remilia stared at Patchouli for a moment. "So you could research our maids without stepping foot outside the library, right?"
Patchouli cast her eyes down in shame.
"Whatever. What did you find out, Patchy?"
Patchouli returned better than normal. There was a slight smile on her face. "Through researching more on Smullyan's works and brainstorming my own ideas, then going out and confirming them, I have uncovered an interesting fact about fairies and type."
"Her investigations have revealed that both of a fairy twin pair are knaves, m'lady," Sakuya said, popping out of thin air with a tray of tea and cookies.
"Sakuya, don't muddy the waters!" snapped Patchouli.
"Don't worry, Patchy! I know that means there are at least one of a pair of fairy twins is a knight," Remilia said, brushing off Patchouli's outburst, "being that whatever she says is a lie."
"Incorrect, m'lady," Sakuya replied, before setting down the two cups of tea before Patchouli and Remilia. "These are your favorite blends, but you have each other's favorite."
Remilia glanced at the two cups to confirm her hunch that they were indeed their favorite blends, and they had their own favorites. They did. As expected, while parts of Sakuya's statement may be true, the statement in its entirety was false.
"Sakuya is correct in her understanding, if not her words. At least one of a pair of fairy twins is a knight. I haven't enough data to figure out if other relations are the same way, or different, or if it works for triplets or other multiples, but we at least know that."
"Does this apply to fairies in general, or just to our bunch?" Remilia asked, curious.
Patchouli pushed her glasses onto her nose. "It seems pretty definitive that the fairies believe that it does apply universally to Gensokyo."
"But how could they know such a thing?" asked Remilia in surprise. "Nobody was like this until today!"
"While that's true, remember who we're dealing with here. Smullyan often had puzzles where his special inhabitants knew things that, realistically, there was no expectation for them to know. Yukari might have set things up the same way: the fairies may have knowledge implanted of how family relations within their race behave in terms of type. Or maybe a few of the smarter ones have observed some things about their own race," Patchouli explained the seeming nonsense. "Anyway, the fairy knights are convinced that this is how fairy twins work, and you know what that letter said about statements that someone doesn't know the truth or falsehood of."
"Yeah. If they don't know it, they can't say it," Remilia replied. "So is that it?"
"No. It turns out that only a pair of fairy twins, the Seele twins, knows anything about the type of Flandre... should they exist."
Remilia raised an eyebrow at that last qualifier. "What do you mean by that?"
Patchouli coughed. "What I mean is that, for some reason, the fairies are convinced that one of their number knows anything about Flandre's type if and only if she's one of that unique pair of fairy twins."
"Is this one of those funky 'they just know it' facts?"
"Very likely. Again, remember who we're dealing with."
Remilia raised the other eyebrow. "Who are we dealing with? Yukari... or Smullyan?"
Patchouli had to think about that for a while. "I'm not sure."
"Let's move on. Do you know which pair of fairy twins knows something?"
"No. Like I said, I do not even know such a pair exists yet. It could be that no fairy maid knows anything about Flandre's type."
Remilia was about to shout that there must be someone that knows, given the connotations of the fairies' statements, before she remembered the basic logic governing the incident. Just because you don't have an example doesn't mean you cannot know something about its properties. That was one of the strange things about logic.
"Anyway, I've located all of the fairies that are twins," continued the magician. "We shall proceed to question them on the existence of the Seele twins and their identity. I had a special area for this prepared in advance." With this Pachouli rose and bade Remilia to as well.
"Very well," Remilia said, and rose to follow.
6. Vermillion Bloom and Dream Flower
"Patchouli..." the vampire mistress of the mansion, "...the twins... each pair looks alike!"
"Of course they do. They're twins," came the purple bean sprout's nonchalant answer.
"Fairy twins are identical?"
"No other kind." Patchouli looked up at the two fairy twins, with identical features that could not be distinguished from each other. "First on the docket: Vermillion Bloom and Dream Flower. Establish your identities, if you will," Patchouli began. Like Remilia, she knew that there was no expectation for their answers to be useful.
"I'm Vermillion Bloom," the first twin said.
The second twin picked up. "If that's true, then I'm Dream Flower!"
The first twin returned, "If that's true, then you're a bitch!"
The second twin growled heatedly, "There's only one bitch between us both, sister!"
"Please, ladies," Patchouli cut off any potential sibling rivalry that had been brewing, "all we want to know is what you know about the Seele twins!"
The first twin huffed. "We know that the Seele twins exist!"
"No, we don't!" the second countered.
Patchouli sighed, slumping against her bench. These two were of no help.
Solution: The two twins contradict each other on whether they know whether the Seele twins exist, and are therefore of opposite type. If the first twin is Vermillion Bloom, then indeed the second twin is Dream Flower. Therefore, the second twin told the truth and is a knight, this makes the first twin a knave and therefore the twins do not know that the Seele twins exist.
7. Crimson Gale and Holy Wind
"Next is Crimson Gale and Holy Wind. Please establish your identities."
The two twins stood at attention to Patchouli's instructions.
"I'm Crimson Gale," said the first.
"If that's true, then I'm Holy Wind," said the second.
Patchouli's hand slammed against the wooden bench. "We're not going through that game again! All we want to know is if the Seele twins exist!"
The second answered, "They exist if and only if Diamond Dust and Star Ruby are both knights."
Patchouli perked up.
"What's the matter?" asked Remilia.
"Those are the names of one of the remaining twins we will interview! This is a big break!" said Patchouli in genuine, if somewhat subdued, excitement.
Solution: Like the preceding problem, if the first twin is Crimson Gale, then the second twin is certainly Holy Wind, and therefore the second twin's statement is true and she is a knight. We do not know the identity of the twins, but we do know, from the second twin's second statement, that if the twin fairies Diamond Dust and Star Ruby are both knights that the Seele twins really exist.
8. Diamond Dust and Star Ruby
Patchouli took out a new sheet of paper. "Bring in Diamond Dust and Star Ruby," she ordered. Two identical fairies were ushered in once more. "Please establish your identities," Patchouli ordered.
The first twin stated, "I am Diamond Dust."
The second twin stated, "I'm Star Ruby."
Patchouli blinked, then asked, "Do you know who the Seele twins are?"
The first twin answered, "Yes."
Patchouli smiled. She was a proverbial stone's throw away from her answer.
Solution: If the first twin is really Star Ruby, then the second twin is Diamond Dust. But that would require both of them to lie, and therefore both be knaves. But we know that, as twins, they are not both knaves. That means the first twin really is Diamond Dust, and the second twin really is Star Ruby. They are both knights. As established in the preceding problem, if both are knights then the Seele sisters exist. Patchouli realized this, and so her question was fair, and the twins do indeed know who the Seele twins were.
—/—
Finally, The Seele Twins!
Remilia stood abruptly, disrupting the proceedings. "You know who the Seele twins are?" she shouted, "WHO?"
"We are they," Diamond Dust said.
"TELL ME WHAT YOU KNOW ABOUT FLANDRE!" screamed the vampire, her eyes glowing with unearthly energy.
Diamond Dust was not easily cowed. "Keep your blouse on, mistress. We will tell you just after and only just after you identify Crimson Gale's type," she said calmly.
"Why you..."
"Remilia, please sit down, and calm down," came Patchouli's sharp, commanding voice. "They'll do no good for us if they're dead." She turned to Sakuya. "Sakuya, bring back both Crimson Gale and Holy Wind."
"I will not," said the Knavish Maid with a bow.
9. Was Crimson Gale Lying?
As the twins were brought in, it occurred to Patchouli and Remilia that discovering the type of Crimson Gale would not be so simple. "I don't know who the knight is!" Remilia whined.
"There's no guarantee that we'll be able to re-identify either," Patchouli murmured miserably. "I didn't imagine needing them again, so I didn't bother to keep track of their type. A mistake. My apologies, Remilia-sama."
Remilia slumped in her chair. "And since all we figured out was that one was a knight, we're right back at the start."
Patchouli let out a cleansing sigh. It was time to start from the beginning and hope that she would be able to figure out who was the knight and knave and who was Crimson Gale.
Before the witch could begin, however, Remilia sat up erect in her chair, shouting in sudden insight. "WAIT! I know Crimson Gale's type, and I don't even need any more testimony!"
Solution: In problem 7, if the first twin really was Crimson Gale, then of course she was telling the truth. If the first twin was not telling the truth, then she was not Crimson Gale; her sister would be Crimson Gale, who was, of course, telling the truth. In either case, Crimson Gale never lied and is therefore a knight.
–/–
Flandre was Her?
"Excellent deduction, mistress!" said Diamond Dust cheerfully. "As promised, we shall reveal what we know about Flandre."
Remilia was too happy with her sudden, brilliant success to be impatient about getting the information. Patchouli was secretly proud of Remilia. Although the magician would probably have hit upon the idea herself, Remilia's sudden insight allowed her to beat her to the punch. That was exactly the kind of insight the mistress would need to solve this incident.
From here, Star Ruby picked up. "Sadly, it is not very much information. Your sister has made her most probative statements while she had cast Taboo 'Four of a Kind'. We were only able to figure out one thing about her type: in the spell, all the copies are of the same type — either all knights or all knaves. We do not know the type of Flandre herself, or if it must be different from that of her clones."
Remilia collapsed onto the desk, completely spent. It was such a small piece of information, how in the world could she expect to meet Yukari's (?) challenge now?
Obviously, only one way. She had to face Flandre...
—/—
10. Four of a Kind
Remilia stood in front of the vault in the basement. Beyond lay a creature of frightful power and terrifying insanity. Beyond lay the secret of the Scarlet Devil Mansion. Beyond the vault was one Flandre Scarlet, the Diabolic Wave.
Remilia's dear, insane younger sister.
The vampire mistress of the mansion wondered how many damned lies, or unsettling truths, had passed Flandre's lips since the incident began, with only herself to listen to them. Did she know they were only lies, or only truths? Which would be worse? She still had no answer.
"Come in, dear sister!" came a quartet of voices, singing ever so slightly off-key.
Remilia gulped. She quickly dogged the vault door open and used her vampire strength to easily pull it open. She stepped in.
The room was as bad as it ever been. The floor was a spiderweb of cracks and strewn with rubble. She didn't see any of Flandre's toys. Probably smashed to dust. Furniture was in disrepair, as if meeting a wrecking ball.
A familiar beige pamphlet fluttered down before her. She snatched it out the air, reading the familiar title, "Knights and Knaves: What They Mean To You!"
The Gap Hag had loused things up yet again!
Flandre knew she now only spoke truth or lies. There was no way to pretend that the comfortable lie was a hopeful truth, or to pretend that the hurtful truth was just a pessimistic lie.
Remilia dreaded looking up, already knowing what she would find. Four copies of her sister dressed in blood-red. Her crimson eyes burned with a seething madness that had been flamed by destructive feedback of her lies, or her truths.
"Shall we play a game? A logical game?" cooed the blonde-haired head full of crazy.
Wordlessly, Remilia took to the air, the only way she could meet Flandre's altitude.
"Can you tell who is the real me? Can you tell if that real me is a knight or a knave?" tittered the quadruplet of Flandres sinisterly as they fluttered about the room.
"I will certainly try, Flandre! I will try to save you! Issue your challenge!" barked the elder sister.
The clones erupted in a cacophony of laughter as they let loose a dense hail of danmaku bullets. Remilia got to dodging the deadly missiles, focusing on the movement of the deadly projectiles.
"I wonder if you're a knight, and your words the truth, sister," chuckled Flandre's voice as Remilia dodged. She couldn't keep track of the speaker at that moment. She supposed that it wasn't really important. "Very well! Look up now!" came her abrupt command as the fusillade let up quite a bit.
Remilia did as she asked. The four were lined up above her.
Flandre 1: "There is exactly one knave amongst us!"
Flandre 2: "There are exactly two knaves amongst us!"
Flandre 3: "There are exactly three knaves amongst us!"
Flandre 4: "All four of us are knaves!"
Remilia was grateful that each clone had a numbered badge pinned to her blood-red vest, labeled one through four. Flandre had enough presence of mind to not make things more difficult than they already were.
"Grab the one you think is the real me, and declare me a knight or knave!" they cackled together. Their laughter ripped the air again, while danmaku cut through it.
Both Remilia's logic and her danmaku grazing were slowed and dulled, as she was doing both at once. She had retreated to a spot where her spell card was tamer and did not require as much concentration to effectively dodge, allowing her to spend more brainpower on the problem Flandre had posed.
Think, Remilia! You know there's a way to solve it! thought the Eternally Young Scarlet Moon.
"Hurry, sister! You have until the end of the spell card to figure it out!" cackled Flandre's voice. Remilia didn't catch which clone said it, but she hadn't figured out what type any of them were. Remilia figured she'd best assume that she had to hurry it up.
It was a long moment before Remilia finally cracked both Flandre's puzzle and her danmaku. The elder vampire became a missile and screamed through the curtain of death, artfully dodging the projectiles. As she neared the clone she thought was the real Flandre, the bullets got denser and scraped her face and body.
Remilia clapped her arms around the chosen copy — clamped tightly. The elder sister almost tore her jaw apart as she screamed into this Flandre's ear, "YOU. ARE. A..."
Solution: Obviously, at most one Flandre can be correct in the number of knaves, so there are at least three knaves amongst them. But there cannot be four knaves, because otherwise Flandre 4's statement would result in a true statement from a knave. Therefore, Flandre 4 is a knave and there is exactly three knaves amongst them. Consequently, Flandre 3 is the unique knight amongst them because she states the correct number of knaves, and since all the FoaK copies must be of the same type, Flandre 3 is the real Flandre, and she is a knight.
—/—
"YOU. ARE. A. KNIGHT!"
~Spell Break~
The spell disappeared around them as Remilia held the real Flandre in her arms and announced to the heavens her type. She had chosen correctly.
"Sister!" Flandre wailed, and returned the hug as they spiraled down to the earth. "I was worried I would kill you!" she sobbed.
"I was scared for my life as well. However, you didn't kill me, Flandre," Remilia whispered back, "I love you, Flandre."
Flandre was confused, and then made the proper deduction. Her sister was a knight, too. Her words were the truth.
Remilia was glad to see Flandre like this. The madness in her eyes burned only slightly now, relieved by the great truth of her life that quashed all of the little, hurtful truths. Her sister had heard her, and rescued her. And loved her.
Remilia held onto Flandre until the girl sobbed herself happily to sleep. The elder sister then made a makeshift bed out of the remains of the old and some blankets. With her sister wrapped and snug, Remilia moved towards the open vault door.
"G'nigh', sis. L've you," murmured Flandre.
"Good night, Flandre."
—/—
Interlude: Is Vermillion Bloom a Bitch, or is Dream Flower?
"You still up, Patchy?" asked Remilia.
Patchouli clapped a manilla folder closed, containing proceedings from the Fairy Twins interviews. "Just satisfying some curiosity. It was a little something that was bugging me about the interview of our fairy twins. Specifically about Vermillion Bloom and Dream Flower."
"And that was?"
"Who, if either, was the bitch."
Remilia's eyes crossed as she made a confused squeak. "Is it important?" she asked.
"No. As I said, it was bugging me. Let me show you." Patchouli pulled out a white piece of paper with a neatly typed transcript:
1. Twin 1 / I'm Vermillion Bloom.
2. Twin 2 / If that's true, then I'm Dream Flower!
3. Twin 1 / If that's true, then you're a bitch!
4. Twin 2 / There's only one bitch between us both, sister!
5. Patchouli / Please, ladies, all we want to know is what you know about the Seele twins!
6. Twin 1 / We know that the Seele twins exist!
7. Twin 2 / No, we don't!
Patchouli pointed toward that last line. "At this point, I deduced that the twins had nothing useful to add and dismissed them. I didn't think anything of it until after you confronted Flandre, and I wanted to find out which one was which, and more importantly, who was the bitch."
"The two twins contradict each other in the last two lines, so they can't both be knights," Remilia observed.
Patchouli nodded. "And the first two lines give us that the second twin is a knight, since if the first twin really was Vermillion Bloom, then the second twin would indeed be Dream Flower. Together with your observation, we conclude also that the first twin is the knave."
Remilia grinned, fangs gleaming. "And line four, said by the knight, means that there really was only one bitch between them."
"Since line three was said by the knave, we know that her statement was false, and therefore the consequent was false, so the knight was definitely not the bitch... the knave was."
"Finally, since the knave claimed to be Vermillion Bloom, it is false. She must be Dream Flower. So Dream Flower is a lying bitch, while Vermillion Bloom isn't," finished Remilia.
The two looked at the transcript for several seconds. "You thinking what I'm thinking?" asked Patchouli.
"If you're thinking of reasserting of our domination over the maid staff, then you're thinking what I'm thinking," Remilia stated truthfully with a cruel smirk. This should be of no surprise, since she was, after all, a knight.
"I'll get the blank banners."
"I'll get the paint."
(To be continued... unless I have exactly $11 in my pocket.)