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 Smullyan.
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 to find herself unable to lie, and therefore a knight. 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.
Remilia and Sakuya have just recently helped contain an outbreak of a disease amongst the Eientei rabbits that caused knights to become liars and knaves to tell truth. This was complicated by Reisen Udongein Inaba contracting a more serious form of the disease, which turned her into a brutal monster, the Vorpal Bunny. The sub-incident was resolved by the timely creation of the cure by Eirin.
Knights and Knaves:
A Logical Adventure of Gensokyo
by Wyrm
Chapter 6:
In Which Remilia Scarlet Hikes Through the Forest of Insanity
Remilia had been enjoying her mid-evening tea when a sealed envelope fluttered down to settle onto her mob cap. With a growl of annoyance, she snatched it off and looked around for any inerrant gaps in the spacetime continuum, but found none. She hadn't been expecting one, since there was no way Yukari would dally looking through one to see her reaction — or if she did, she would be sure to have it camouflaged.
Mumbling unkind things about the Great Troll of Gensokyo, Remilia ripped open the envelope to read the message within.
Dear Remilia,
Before now, knight or knave, everyone you have encountered had beliefs coincident with reality, and therefore told the truth or lied according to that correct view of reality. Now you must consider that not only the being may be lying, but may not have the correct view of reality in the first place.
Something strange has happened in the Forest of Magic — a peculiar madness has taken hold there, driving half the beings there insane. (I strongly suspect Lady Yukari's involvement this time, but I cannot say for sure.) Every being in this new Forest of Insanity is either sane or insane. A sane being believes only true propositions and does not believe false ones, while an insane being only believes false propositions and does not believe true ones. This is where the caveat from the first message comes in. Every being you have encountered so far is sane, at least insofar as this stipulated definition is concerned. Yes, even the powered-up Shizuha Aki was sane according to this definition: at best, the EX-ercise Machine only distorted her desires, not her beliefs.
Sane or insane, someone cannot both believe some proposition and its negation. However, this does not apply to the logical consequences of these beliefs. Just as a (sane) knave can make logically incoherent statements, an insane person can believe logically incoherent statements.
Sane knights and knaves have the familiar behavior, because their beliefs coincide with reality and can reliably tell the truth or lie about them. Insane knights make claims that are false, because they sincerely believe incorrect propositions and tell the truth about them (an insane knight can claim to be a knave, because she sincerely believes she is a knave, and as a knight tells the truth as she knows it). Insane knaves make claims that are true, because they sincerely believe negations of true propositions, then lie about the negations leading to a true statement (an insane knave can claim to be a knave, because she sincerely believes she is not a knave, but as a knave lies about this mistaken belief).
Natives of the forest are either sane or insane. Strangers to the forest such as yourself, however, are all sane. The cause of the insanity is unknown, but those who leave within a day or so upon entering have a reasonable chance of leaving the forest with their sanity intact. The longer you stay, however, the greater your chance of being driven insane. As your stay will be short, there should be no danger. On the off chance one of you are driven insane, however, I would ask that you write down your observations.
Your objectives this time are Lily White, Letty Whiterock, Kogasa Tatara, Koishi Komeiji, Marisa Kirisame, and Alice Margatroid. You must not only determine the types of these people, but also their sanity — sane or insane.
Be careful in dealing with the Seven Colored Magician. I've been hearing strange rumors about her as of late.
Good Luck and Be Safe.
—Ran Yakumo
"That's not how insanity works, Ran Yakumo! (I should know!)" Remilia declared as she finished reading off the letter aloud. At the time, she was standing in the library, over Patchouli (as best as she could, given her rather diminutive height). It was for her benefit that Remilia read the letter aloud.
Patchy, for her part, had calmly listened to Remilia's recitation of Ran's letter here in Viole. She then stated, matter of factly, "It's a stipulated definition, Lady Remilia. It overrides the usual meaning." Remilia supposed she had a bewildered look on her face, because Patchouli continued, "This definition of 'sane' and 'insane' comes straight out of Smullyan's book, and you should've reached that point by now, had you been keeping up with your reading."
Remilia gave an embarrassed grin. "Eh heh heh. Sorry, and you're right. I should have known that the Gap Hag would be using more of Smullyan's scenarios," she said, laughing off her failure to keep up with her studies. "Speaking of which, do you have any thoughts about that?"
"I have some observations. Ran leaves a lot unsaid in her letter. Implied by the statements she has given, but still, a great deal unsaid that will trap you if you are unwary," Patchouli stated evenly, suddenly taking on the air of a university lecturer.
"Oh?"
"For one thing, Ran hasn't out-and-out said that insanity reverses the truth value of statements made by knights and knaves," the unmoving magician began.
"Isn't that obvious?" asked Remilia.
"Why do you think it is?"
"If someone is insane, then their beliefs are reversed. An insane knight believes she is not knight, because otherwise her belief is correct and the insane cannot believe correct propositions. She must tell the truth as she believes it, and so claims not to be a knight."
Patchouli smiled. "You are correct that an insane knight does not believe she is a knight, because that belief is a correct one and —as an insane being— she cannot believe correct propositions, but I see nothing here that asserts that she does believe she is not a knight. What if she doesn't know either way, unlikely as that is?"
Remilia felt her face flush slightly with embarrassment. "Oh," she acknowledged Patchy's point in a very small voice.
"Furthermore, what if she's an insane knave? An insane knave does not believe she is a knave, because that belief would be correct and —being insane— cannot believe it," Patchouli continued, "but that doesn't mean she believes she is not a knave. See the problem now?"
"Yeah, I do," Remilia said with a frown. She had still much to learn about logic.
The magician continued her impromptu logic lesson. "Instead, we must go to one of the previous missives. Ran said that, in order to make a statement, the person must know the truth of what she's saying. She must know whether her statement is correct or incorrect — implying that she either knows the statement to be true, or knows the statement to be false. However, she has also stated that what I have put forward is a simplification of the actual situation.
"Everyone you have interviewed so far is sane. That is, their beliefs match with reality. So if we may refine that rule, it means that a person cannot make a statement unless they either believe the statement, or its negation. This collapses to Ran's simplified statement in the case of the sane people you've encountered thus far, because we may take belief in truth as knowing."
A light seemed to turn on in the loli vampire's head. "Ah. So we must actually wait until the person talks about something before we can conclude that they believe either way," she said in realization.
"Yes. Now, for a matter of externalities, like whether one is a knight or knave, what can we conclude about some statement, S, when said by either an insane knight or an insane knave?"
"Well..." Remilia hesitated as she thought it over. "An insane knight who says S seriously believes S is true, but being insane, S must in fact be false, so whatever an insane knight says is false. An insane knave who says S is lying about her belief in S, so she must seriously believe that not-S is true, but being insane, not-S is false, thus S is true, and therefore anything an insane knave says is the truth."
Patchouli smiled at her friend and student. "Spot on, Lady Remilia," complemented the librarian. "What you say is true for direct statements of fact, like whether one is insane or a knave, but what about statements of one's beliefs?"
"Um..." Remilia thought about this, but quickly got tangled on the beliefs of beliefs. "Oh dear, that might get complicated."
"Your caution will serve you well, I believe."
"Patchy, did you just tell a joke?"
Patchouli blinked, eyes slightly widening at the observation. Mentally reviewing her last statement, she supposed that, yes indeed, she had told a joke. "Not intentionally, and it's a very weak one at that," she admitted. "But back to the discussion. There may be some category of creatures where it is true that if he or she does not believe some proposition S, then they do indeed believe the proposition not-S. We may call such creatures opinionated on S, and creatures who are opinionated on every S... simply 'opinionated.'"
Remilia remembered how Patchouli's handedness terminology caught on without having needed time to spread around. "I suspect that your 'opinionated' term is going to become very in-vogue by the time we arrive at the Forest," the lady of the house said dryly.
A ghost of a smirk crossed Patchouli's lips.
—/—
Fruitcake Fairies in the Forest
The Forest of Magic loomed ahead of the two travelers as dusk turned to night. Remilia closed her parasol that she used to shield herself from the sun's deadly rays as she traveled, as here in the forest, it was unneeded. This was not only due to the fact that it was night, but also because of the fact that the tree's healthy canopy would shield her in case they were caught by the sunrise.
"Well, let's get started," announced Remilia.
"It will not be immediately," lied Sakuya. "Where shall we start, mistress?"
"That thieving black-white ordinary human magician, with a name sounding like 'Melsia Beriberi,'" the young vampire declared.
"Her name is not Marisa Kirisame."
61. Lily White is Our Guide?
A white-clad fairy with a conical hat with red zigzag accents fluttered down on downy wings. She held up her hand in greeting, "Hello! Fancy meeting you again, Sakuya Izayoi!"
Sakuya jumped. She recognized this particular fairy! "Lily White? You're not here!" she gasped, then clamped a hand to her mouth as she blushed in embarrassment.
The Fairy Herald of Spring Lily White giggled, as Sakuya's statement had just exposed her as a knave, but said nothing. The Fairy of Spring sobered, opening a piece of paper she had been carrying. Remilia idly noted that an envelope was still safety-pinned to her clothes. "I got a message from the Great Troll's shikigami that I am to be your guide to the forest," she said, waving the letter about too quickly for anyone to read. She frowned at the letter, studying it intensely. "Don't know why she did that. It's not as if I'm completely up on everything that happens here."
"Are you sane, Lily White?" inquired the vampire.
Lily huffed indignantly, putting her tiny, white-bedecked arms on her tiny, white-bedecked hips. "Of course I am! I'm not a sane native of the forest, though."
Remilia knew then that Lily White could be trusted to guide them.
Solution: Lily White claimed that she is sane. If she were a knave, then she actually believes she is not sane. If she were sane, then her belief is incorrect because she's actually sane and is not not sane, and sane people cannot believe incorrect propositions. But if she were insane, then her belief is correct because, being insane, she is indeed not sane, and insane people can only believe incorrect propositions. The assumption that Lily White is a knave leads us to a contradiction. Therefore, Lily White is not a knave. She is a knight.
Because she is a knight and claims that she is not a sane native of the forest, she believes it. If she is insane, then she is correct in her belief that she is not a sane native of the forest, because she is an insane native of the forest. But this means her belief is true, and being insane she cannot believe true beliefs. Therefore, Lily is sane. Thus, her belief of not being a sane native of the forest is true. Since she isn't an insane native of the forest either, she is not a native of the forest; she is a stranger to the forest (and sane).
—/—
"You're not from around here, are you?" Remilia observed.
Lily nodded curtly, but with a smile on her face. "Correct. In the off-season, I live just beyond the forest, quietly minding my own business and not attracting attention to myself," she said.
Remilia glared at her. "Considering all you do during the break of winter is draw attention to yourself, that sounds mighty strange coming from you."
"Don't confuse my job with my private life, Remilia Scarlet," Lily huffed indignantly. "Anyway, I didn't get hit with whatever magic drove many in the forest insane. All I felt was the wave of excess magic shucking off the forest," the fairy continued, "I went into it to find all my friends and find out whatever I could about what happened to them. I found in addition to the usual knight/knave shenanigans, this layer of insanity." Lily's face fell, a worried look settling about her. "It was so very difficult to deal with. I only managed to find out a few things about my friends, but sadly, there's still a bunch I don't know."
Lily White brightened as she held up her letter from Ran. "Ran said that you'll help me find out what happened to my friends! Thank you in advance, Lady Scarlet!"
Remilia felt slightly nervous. True, in solving these puzzles, Lily White would learn what happened to her friends, but it was apparent that Ran had told her an incomplete story. Helping her would be a side-product, not the main objective.
Lily continued. "I have quite a few pairs of twins that Ran says you'll help me with. The twins' names are as follows: Clearwater, Dewweed, Jasperstone, Humblebee, Gallflower, and Tumbledrum. Ran says that the sane knights of these fairies should be able to help you." She paused, studying the letter intently. "Um..."
"What is it, Lily?" asked the vampire youth.
"Lady Scarlet, what does 'opinionated' mean?" Lily asked, a confused look on her cute little face.
"What?"
"Ran's letter says that all of the fairies I just mentioned are 'opinionated,'" Lily clarified. "Do you have any idea what that means?"
Remilia let out an exasperated sigh and began explaining, not only the meaning of that particular word, but of their situation in general...
—/ Girls are Explaining... /—
"Un. I see! So either Lady Yakumo or her shikigami has set various tasks in front of you to solve this mess!" Lily White said in understanding. "Then Ran did not tell me to guide you just so that you could help me find out what happened to my friends, as I had assumed."
"Very astute of you, Herald of Spring."
"Heh heh!" Lily giggled, looking embarrassed and proud at the same time. She snapped out of this and a determined look settled upon her features, steeling herself to the task ahead. "Very well! If helping you helps me, then I'll be glad to do so! I shall tell you what I have learned about my fellow fairies!" she said boldly, staring out with the first fact she had learned: "In the forest, both of a pair of a fairy twins cannot both be knights."
"Outside the Forest, fairies of such a pair cannot both be knaves," countered Remilia. "Is that still true?"
Lily nodded vigorously. "Yes. So, a pair of fairy twins is composed of a knight and a knave — at least, when they're natives of this forest."
"This actually might make things easier," Remilia mused. After all, there would be only two possibilities to consider instead of three.
Lily paused to read the letter further. "It says here that for some of the fairy twins, the knight will be sane. We are to bring along the sane knights, and will help us in the next phase of your journey," Lily explained.
Remilia frowned at that. "Seems the Gap Hag chained the problems again," mumbled the vampire sourly as they approached the first pair of fairy twins.
62. Onyx and Beryl Clearwater
"These two are Onyx and Beryl Clearwater, and they're twins," Lily said, gesturing at each of the identical twins in their turn. Unlike in the Scarlet Devil Mansion, each fairy was wearing a dress that suited their own taste, and so could be used as a means to distinguish them. "I'm afraid I don't know anything else relevant to your quest about them," Lily continued, somewhat subdued from her original introduction.
"It doesn't sound like you really tried."
"I'm just a fairy! Cut me some slack!" whined the fairy herald.
Shaking her head, Remilia confronted the twinned fairies. "Onyx, Beryl, would you mind telling me about yourselves?"
One twin, Onyx, replied first. "We are both insane."
Beryl, the second twin, countered the first, "That's not true!"
"I am a knight," Onyx added.
Remilia knew then both their types and sanities.
Solution: Onyx's statement is either true or false. If it is true, then Onyx is insane, and the only insane inhabitants of the Forest who speak truly are knaves. Thus, Onyx is the knave and Beryl is the knight (both insane). If Onyx's statement is false, then at least one of them is sane. If Onyx is sane, then she has lied, and so she is the knave, and Beryl is the knight. If Beryl is sane, then has spoken truly in contradicting Onyx, and so must be the knight, making Onyx the knave (and sane). In all cases, Onyx is the knave and Beryl is the knight.
Onyx is a knave, so by her second statement, Onyx doesn't believe she is a knight, and being opinionated, she therefore believes she is not a knight. This is true, for she is a knave. Therefore, she believes something that is actually true, and she is thus sane. Beryl is a knight, and therefore she really believes that the twins are not both insane, and of course, we have just shown that Onyx is indeed sane, so her belief that they are not both insane is true. Because Beryl believes a true belief, she is also sane.
63. Honey and Faith Humblebee
Remilia directed Beryl Clearwater to follow them, as she bore information for the next part of Remilia and Sakuya's quest.
The tour of four persons proceeded through the forest, Lily White leading the way, until they happened upon another pair of fairy twins. Lily smiled at the sight of them. "These are Honey and Faith Humblebee. I know nothing about them relevant to your quest, either."
Remilia let out a sigh of exasperation. "Fine. Honey and Faith, tell me something about yourselves."
"I am a knight," said Honey.
"I am a knight, too," said Faith.
Honey nodded towards Faith. "My sister is sane," she claimed.
It didn't take long for Remilia to figure out their sanities and types.
Solution: Suppose Honey really is a knight. Then she believes she is a knight, and because she actually is a knight, she is sane. If Honey is a knave instead, then she believes she isn't a knight and is lying about it. But she actually is not a knight, and so is also sane. Therefore, Honey is sane. The same argument applies to Faith, and she is also sane.
Honey believes either that Faith is sane or Faith is insane. But of course, Faith is sane, as is Honey, and therefore Honey cannot believe Faith is insane, and must believe she is sane. Since this is exactly what she claims, Honey is a knight, and therefore Faith is a knave.
64. Lyra and Bonnie Dewweed
Honey Humblebee joined Beryl Clearwater on their troupe as hangers on.
A bit further into the forest, Lily stopped and pointed towards another pair. "Here are Lyra and Bonnie Dewweed. Like the others, I haven't figured out their sanities."
"If you will..." Remilia prompted.
"I am a knave," said Lyra.
"I am a knight," Bonnie said.
"Bonnie and I are the same as far as our sanity goes," Lyra added.
Remilia knew then she was wasting her time with these two.
Solution: Lyra claims to be a knave, which makes her insane. For a knight claiming to be a knave must believe she is a knave, which is not true and thus makes her insane, and a knave that makes such a claim must believe she is not a knave, which is again not true and thus makes her insane. Bonnie claims to be a knight, which makes her sane (as we found in the previous problem, a person claiming to be a knight is sane). Therefore, the two are not alike as far as their sanity goes.
This contradicts Lyra's seconds statement: contrary to her claim, Lyra and Bonnie do not match in sanity. Since Lyra is insane and uttered an untruth, she must be the knight. This makes Bonnie the knave, who is also sane.
65. Ruby and Amber Jasperstone
Neither Lyra nor Bonnie Dewweed joined the troupe, being unsuitable for Remilia's purposes. A few minutes further and they came upon yet another pair of twins.
As Lily spied this third pair, she clapped her hands together with excitement. "Ah! This pair I know something about!" she said happily, "These are the fairy twins Ruby and Amber Jasperstone! I managed to figure out that one of them is sane and the other is insane!" Lily sounded quite proud of herself.
Remilia scratched the back of her head, completely disinterested in Lily's self-congratulation. "So," she said, addressing the two fairies, "Do you want to tell me about yourselves?"
"My sister is a knave," Ruby said, pointing at Amber.
"My sister is insane!" Amber roared.
Remilia was about to leave them behind on sheer principle, but she found a better justification.
Solution: Suppose Amber really is a knave. Then Ruby is the knight, and furthermore, must be sane, as she's told the truth. This means that Amber, being a knave, has lied and actually believes her sister to be sane, which is true. But this implies that the both of them are sane, which puts lie to Lily's statement, and we already know she is a sane knight. Therefore, Amber cannot be the knave; she is the knight.
This makes Ruby's statement false and makes her the knave. Therefore, she doesn't really believe that Amber is a knave, and being opinionated she must believe that Amber is not a knave, which is true: Amber is a knight. Therefore, Ruby is sane. But Amber claimed that Ruby is insane, and she is a knight, therefore she really believes Ruby is insane, which is wrong, and therefore Amber herself is insane.
66. Viola and Rosie Tumbledrum
Again, no one joined the train of fairies.
"We're picking up fairies very quickly," observed Sakuya in her knavish manner.
"Indeed, we are not. With any luck, that will make any puzzle connected with getting all of them a snap," Remilia speculated. "Speaking of which, it looks like we're coming up on the next pair."
"This is Viola and Rosie Tumbledrum!" Lily said enthusiastically, "Like the Jasperstone twins, one of the Tumbledrum twins is sane and the other is insane." The fairy turned to the twins, and instructed them, "Say something about yourselves, girls!"
"I am a knave," Viola boldly claimed.
"If that were true, then I am a knight," Rosie added.
Remilia considered this one easy.
Solution: Since the two fairies differ in both type and sanity, they either both tell truths or both falsehoods. For if the knight is sane, the knave is insane, and thus both tell truths, and if the knight is insane, the knave is sane, and so both tell untruths.
If Viola really is the knave, then Rosie is most certainly the knight, which means that Rosie's statement is true. Since Rosie tells truths, by the above derivation, they both must do so, so the knight must be sane and the knave insane. Viola is definitely insane — no sane being can claim to be a knave. Therefore, she is the insane knave and Rosie is the sane knight.
67. Petal and Posey Gallflower
Rosie Tumbledrum joined Honey Humblebee and Beryl Clearwater, being suitable to Remilia's future task.
"These are Petal and Posey Gallflower," said Lily, introducing the twins.
"You wouldn't happen to know anything about them, would you?" the vampire asked the spring herald.
Lily, predictably, shrugged. "Sorry. I don't know anything further about them," she admitted. "Please tell us something about yourselves."
Petal started. "I am either a knave, or a native."
"Petal is an outsider. She is also a knave," Posey contradicted.
Lily clapped her hands together in a flash of realization. "Hey! I figured it out!" she shouted.
Remilia gaped. "What? How did you—?"
"I haven't just been floating around being a convenient plot device, you know! I've been learning!" Lily said with a grin.
Solution: If Petal's statement is false, then she is neither a knave nor a native. She must then be an outsider knight, and if she is an outsider, she is sane, so she is a sane knight. But sane knights tell the truth, so her statement cannot be a lie. Therefore, she is either a native or a knave (or both). Because the statement is the truth, Petal is either a sane knight or an insane knave. If she is a sane knight then she is a native because she is not a knave; if she is an insane knave, then she is a native because she is insane.
Posey's first statement is false, because Petal is indeed a native — Posey has lied or believes one. The twins therefore must be of equal sanity (see previous puzzle). Since they do not both speak the truth or speak wrongly, but are definitely of different type, they cannot have different sanities.
Posey's second statement is also false, because Posey has established herself as someone who tells untruth. Therefore, Petal is a knight, and therefore a sane knight because she tells the truth. This establishes Posey as a sane knave, because the two have equal sanity, and opposite type.
—/—
More Fairy Family Relationships ~ Fairy Cousins
Petal Gallflower was the final addition to the entourage now consisting of herself, Rosie Tumbledrum, Honey Humblebee, and Beryl Clearwater. All of them were sane knights.
Remilia confronted the four fairies. "Alright, you four. You all are sane knights. What is this mysterious help that Ran Yakumo expects you to provide me?"
Beryl Clearwater spoke first. "There are are four pairs of fairy cousins we must find: Mary and Gertrude Pennypease, Carol and Tammy Dandelion, Penelope and Catherine Caneriver, and Halley and Jade Dallywinds," she said. "We each have one fact to reveal about these pairs of fairy cousins. My fact is that each sane fairy from these pairs has a key that will be important later and will be given to you upon request, although we are not allowed to tell you much more about what the keys are or what they do."
"Fairies have cousins?" Remilia asked, her surprise catching her off guard.
"Yep, they do," nodded Lily White, the fairy knight.
Remilia narrowed her eyes. "And I assume that being cousins implies something about they being knights and knaves, or sanity or insanity?" prompted the vampire.
Lily nodded again. "Yep. In this forest, fairy cousins are the same type: they are either both knights, or both knaves."
Remilia realized immediately what that meant. "More pointless puzzles. This is just blatant busywork."
68. Mary and Gertrude Pennypease
Lily flew up to two pink-haired fairies, a small one with a chubby face, and the other tall with a thin, regal face. "These two are Mary and Gertrude Pennypease!"
"There is nothing I need tell you about those two," Beryl added.
"Typical. Let's get this over with," Remilia grumbled. "You two. Tell me about yourselves."
Mary, the chubby-faced one, said, "My cousin is a knave."
Gertrude, the tall, regal faced one, said, "My cousin is a knight."
Mary added, "One of us is sane and one is not."
Remilia figured out who held the key and collected it.
Solution: Exactly one of the two cousins must have told the truth, because they are either both knights or both knaves. They cannot have equal sanity because then they would match in their ability to tell the truth, which they don't. Thus, Mary is telling the truth, and Gertrude an untruth. Therefore, they are both knaves. Since Mary is a knave who tells the truth, she is insane. Thus, her sister is a sane knave, and has one of the keys.
69. Carol and Tammy Dandelion
Remilia regarded the gold (or perhaps brass) key with a diamond shaped bow — 'diamond' as in the card suit, and the bow even had a red gem that might have been a ruby in its center. It looked like an ordinary ward lock key, but knowing its probable source, it probably unlocked some cosmic horror, or maybe a tea chest full of really tasty tea.
"No pair of fairies in sight," quipped Sakuya, then pointed towards the obvious pair of fairies right ahead of them.
Lily buzzed up to the pair. "These are Carol and Tammy Dandelion!" she said.
Rosie Tumbledrum pulled up besides Lily White. "I can tell you something of these two: the both of them are opinionated," she said, breaking into Lily White's introduction.
The herald of spring was pretty put out by this interruption. "Hey! That's rude!" she squeaked angrily, windmilling her arms.
"Yes, it was," Rosie said with a smile.
Of course they'd agree, thought Remilia, They're both sane knights.
The vampire turned to the two. "Enough of this tomfoolery. Tell me something about yourselves, you two," she curtly ordered the fairies.
"Whatever my cousin says is true," said Carol.
"My cousin is insane," added Tammy.
Remilia puzzled on that for a minute before asking, "Tammy, do you believe you are sane?"
"Yes," came her answer.
Remilia collected all the keys the two had.
Solution: Suppose Tammy really does tell the truth. Then Carol is either a sane knight or an insane knave. But if Tammy tells the truth, then Carol is insane. Therefore, Carol is an insane knave. Suppose Tammy tells falsehoods. Then Carol is either an insane knight or a sane knave because her statement is incorrect, and because Tammy tells falsehoods, her claim means that Carol is actually sane, and because she tells lies Carol must be a sane knave. Therefore, depending on whether Tammy really is telling the truth or not, Carol is either a sane or insane knave, and Tammy is also a knave of the same sanity level.
By her answer to Remilia's question, Tammy believes she does not believe she is sane. This implies that she is insane. For if Tammy is sane, then she really doesn't believe she is sane. Because she is opinionated, Tammy then believes that she is not sane. But this is not correct: she really is sane, and she believes a falsehood. Thus, if she is sane, she is insane. Therefore, Tammy must be insane. This makes her an insane knave, and therefore she really does believe she is sane (but of course, this belief is wrong, being insane). Because Tammy is insane, Carol is too.
Because each of them are insane, neither has a key.
70. Penelope and Catherine Caneriver
"Lady Remilia, look!" Lily shouted as she dragged two fairies to the group. "These are my best friends, Penelope and Catherine Caneriver," bubbled the Fairy Herald of Spring, introducing the two in turn. "Tell Lady Remilia something about yourselves, girls!"
Penelope, a fairy with pink hair, said, "We are both insane."
Catherine, with blue hair, nodded in agreement. "That is true."
"I understand," Remilia replied. "Are you a knave, Penelope?"
"Only if my sister is as well!"
Their statements revealed to Remilia who of the two had a key.
Solution: The two cousins agree that the both of them are insane, and since they share the same type, they must also share the same sanity: if one is insane, both are insane. If they are both sane, then they cannot believe they are both insane, because that belief is false. If they are both insane, then they cannot believe they are both insane, because that belief is true. Therefore, neither believes they are both insane. Penelope and Catherine therefore believe the opposite of their claim and are therefore both knaves.
Penelope claimed that she was a knave only if her sister is. Of course, they are cousins, so if Penelope is a knave, then Catherine is also a knave, so her statement is true. She is thus a knave who tells the truth — an insane knave. And of course, if she is insane, Catherine is too. And since they're both insane, neither was in possession of a key.
71. Halley and Jade Dallywinds
"May I introduce to you Halley and Jade Dallywinds!" said Lily, throwing out her arms with joy. Remilia shook her head.
"You will now tell us something about yourselves," Beryl said to the cousins. It was not a command, but a statement of fact.
Halley, a fairy with light brown hair, said, "At least one of us is insane."
Jade, with light green hair, retorted, "That's not true!"
"We are both knights," Hally added.
Remilia figured out who had a key and collected them.
Solution: Halley and Jade disagree with each other, and because they are the same type, they must be differing in sanity: one is sane, the other is insane. This means Halley is telling the truth, so she is telling the truth when she says they are both knights. Therefore, Halley is a sane knight and has a key, and Jade is an insane knight.
72. Pipi
Remilia took the ward lock key with the heart-shaped bow from Halley and continued on their way. She noted that the heart key had a red gem in its bow like the diamond key.
The next fairy they encountered was a small, lone fairy, easily the smallest Remilia had ever seen. (Not that she'd seen many. Remilia would have to ask Reimu about fairy sizes one day.) She was unaccompanied.
Suddenly, the fairy shouted, "I believe that I have both a cousin and a twin!"
Remilia was taken aback by Pipi's sudden outburst. "What?" she asked, but the fairy had already skedaddled away.
Lily smiled gently. "Pipi is rather shy," she said.
Honey laughed, "I also know Pipi. She is a native of the forest and opinionated on that particular belief."
Remilia had figured out one thing about Pipi.
Solution: Pipi is a knave.
Pipi could not have both a cousin and a twin. For she would be of opposite type to her twin, and her cousin would be the same type as her, and at the same time be the same type as her twin (because her cousin is cousin to her twin as well as to herself). Thus, the fairy's twin would be of both the same type and of opposite type to herself, which is impossible. Therefore, Pipi (or any fairy of the forest) cannot have both a cousin and a twin.
Suppose Pipi really does believe that she has both a cousin and a twin. Then she has told the truth, because this is exactly her statement. On the other hand, she must be insane, because this belief she holds is surely false. The only individuals in the forest that are insane and speak true statements are insane knaves.
On the other hand, suppose Pipi doesn't believe that statement. Since she is opinionated on whether she has a cousin and twin, she must believe that it is not the case. Since it cannot possibly be the case, her belief is true and she is sane. She cannot believe that she does believe the statement, because she in fact does not believe it and sane people do not believe falsehoods. But she has made a statement about it, so she in fact believes she does not believe the statement. Again, she has lied about her belief, so Pipi must be a knave! We cannot tell if Pipi is sane or insane, however.
73. Pinkie Applebloom
A fairy wearing a pink dress and having a head full of pouffy maroon hair wove her way erratically through the trees and the flowers like a drunken bumblebee. She was so atypical that Remilia would have bet money that she was involved in their quest.
"Do you know anything about this fairy?" she asked Lily, pointing at the unusual fairy.
"She goes by the name Pinkie Applebloom, and she's a strange one."
"That's a matter of opinion," Remilia pointed out.
Lily exploded, flapping her arms in a tizzy of excitement, as she hastily explained why Pinkie was a nut (though not necessarily insane). "You don't understand! She's totally random! Even before the current incident, she was an enigma filling, wrapped up in perplexity bread, sauteed in a bewilderment sauce and garnished with imponderable parsley, served with a side of conundrum."
"Surely, you're exaggerating," said flatly. The logic sense in her nagged, however. Sane knights like Lily were not prone to exaggeration, as it was a form of lying. Nevertheless, she ignored it and addressed Pinkie Applebloom directly. "Pinkie, please tell me about yourself."
"I believe that I am a knave!" Pinkie announced, quite loudly. She also struck a pose that would not look unusual on Iku Nagae.
Disco feevahhh! a rogue thought floated through Remilia's mind, as she searched for a way to answer her. "...I see. Pinkie, are you—"
Pinkie cut her off. "It is not the case that I am opinionated if I am a knight. Also, I have a key." With that, the fairy flew off, leaving a bewildered troupe of Gensokyoites behind.
Lily grabbed her head in frustration. "Grr! What did I tell you! Completely random! I didn't figure out anything about her!" she raved. The fluttering of her wings increased and she began to ascend. "I'll chase her down!"
Remilia casually plucked her out of the air before the fairy herald could pursue. "On the contrary, Lily, I've figured out everything about her."
Solution: This one is tricky! Believe it or not, Pinkie is a sane knave, and she is unopinionated!
Suppose Pinkie really believes she is a knave. If she's a knight, then she is insane, because she's not really a knave — she is a knight. But insane knights do not tell truths, and this fairy has claimed exactly that she believes she is a knave. Therefore, she cannot be a knight, but a knave. Because she does believe she is a knave, which is correct, she must be sane. But then she has told the truth about her believing she is a knave, which a sane knave cannot do. Therefore, Pinkie cannot really believe she is a knave.
Pinkie's statement is therefore not true; she does not believe she is a knave. She can only be an insane knight or a sane knave. Therefore, her statement that it is not the case that she is opinionated if she is a knight is also not true: if she is a knight, she is indeed opinionated. Therefore, if she is a knight and she doesn't believe she is a knave, then she believes she is not a knave. This is true, so she is sane. But this would make her a sane knight, and she can only claim to believe she is a knave if she is an insane knight or a sane knave and she is neither, so she cannot be a knight at all. Pinkie must be a knave. Since we have already shown that Pinkie must be an insane knight or a sane knave, and she is not a knight, she must be a sane knave.
But if Pinkie is a knave and doesn't believe that she is a knave, then would that imply that she is insane? No, only if she is opinionated (in particular, she is opinionated over whether she is a knave), for which not believing she is a knave is equivalent to believing she is not a knave, which would make her insane. However, she is not necessarily opinionated if she is a knave (only if she is a knight), and if she were opinionated, she could not be a knave either. Therefore, she is unopinionated (in particular, not opinionated on whether she is a knave) and therefore neither believes she is a knave nor believes she is a knight. (She must have missed the memo!)
And finally, since she's a sane knave, she's lying about having a key, so she has no key.
—/—
Party of Four
"And so Pinkie is a sane knave who is unopinionated! That's neat!" Lily exclaimed giddily and clapped as her head finally wrapped completely around Remilia's explanation of how she figured that fact out.
"Thank you," Remilia replied, preening a bit. Sure, the praise of her intellect was from a fairy, a race not known for its smarts, but praise was praise. Remilia realized this quickly and did not preen long, knowing how pathetic it might seem, so she moved onto the next topic, "Those two encounters seem rather out of place. Neither of them had a cousin we could talk to."
"Hmmm..." mumbled the white-bedecked fairy.
A moment passed before Sakuya coughed politely, holding up an envelope. "This letter did not drop from the fairy Pinkie Applebloom's robes," she lied.
"Ah. That's the explanation. Alright, let's see what Ran has to say."
Dear Remilia,
One of your fairy twin companions is the bearer of a secret that will allow you to proceed with your quest. Sadly, only Lily White and but one of your fairy twin companions are allowed to accompany you and Sakuya further. Here you must make a choice of which will accompany you — after you make your choice, the rest of the fairies will disappear, one way or another. It is advisable that you choose the fairy who is the bearer of the secret I have mentioned.
But which to choose? That is the challenge! You are allowed to ask one and only one question of a single fairy twin you have accumulated. You are to make your decision on the basis of her answer. All these fairies will know who the secret-bearer is. However, they have been instructed not to directly reveal any but their own status as the secret-bearer. If you ask the identity the secret-bearer by name or any similar question, they will refuse to answer and you will have wasted your only question.
Petal Gallflower bears two clues that will help you. A little thought will show you that what I have given you, these two clues, and your single question will be sufficient for you to find the secret-bearer.
Good Luck.
—Ran Yakumo
"Confound these puzzles!" Remilia growled.
74. Who has the Secret?
Petal Gallflower flew up to Remilia, knowing what was in the letter, and gave her a few clues. "The first clue mentioned in Ran's letter is that the secret-bearer is an outsider to the forest. The second is that if a fairy is an outsider, so is her twin."
Ran was right. A little thought did give her enough information to formulate a question that would tell her all she needed to know. She directed her question to Beryl Clearwater, and from her answer, Remilia was able to tell who should be her forth companion.
Solution: Many questions will work, but the one Remilia asked of Beryl Clearwater was the simplest: "Are you the secret bearer?"
Petal Gallflower established that fairy twins are either both native to the forest or both outsiders, and that the secret-bearer is an outsider. Furthermore, Petal Gallflower had established herself as a native with her statement in a preceding problem, and Rosie Tumbledrum's sister, Viola, is insane, and therefore a native, so Rosie is also a native. This leaves Beryl Clearwater and Honey Humblebee as the the possible secret bearers, as only they can be outsiders.
Ran's letter says that Remilia's question is within the rules, so Beryl will answer. If Beryl claims to be the secret bearer, then as a sane knight, she is indeed the secret bearer. If she claims no such thing, then she is telling the truth and she is not the secret bearer, which implies that someone else is. The only other possibility is that Honey is the secret bearer.
—/—
Beryl had answered Remilia in the negative; that she was not the secret bearer. Remilia immediately pointed to Honey Humblebee and announced, "I choose you, as you are the secret bearer."
Honey nodded as she bowed to Remilia. "Excellent choice, Lady Scarlet. You have chosen wisely," she said as the other fairies dispersed. Remilia let them go, as Honey was the only one she needed, and she was staying.
"I realized pretty quickly that I should be able to logically exclude two of your compatriots, narrowing the field to you and Beryl," Remilia explained. "Ran's challenges are difficult, but any more possibilities would make the challenge unfair."
Honey nodded with a smile. "Very nice insight. But now we need to find Letty Whiterock, as per your instructions," she said with a smile.
Letty Whiterock was not hard to find. She was sitting on a rather large boulder, in the center of a large cold spot in the forest, humming to herself.
"I would have thought that you and Letty would only appear in different seasons," Remilia remarked to the herald.
Lily frowned. "Who told you that? I'm only the herald of spring, not its essence. I exist in other seasons, too, as I am doing now," the fairy herald said.
Remilia thought back to the Aki sisters they had met just a couple of days ago. "I suppose that's true, given the Aki sisters," she remarked aloud, "But how do you get back to the west without anyone seeing you?"
"Hee hee! Fairies' Secret!"
Remilia felt a touch of annoyance being patronized by a fairy. Still, she tried not to let any annoyance seep into her voice. "That still doesn't answer my question: how can Letty appear out of her season?"
The fairy herald shrugged. "I donno. Maybe, like the heralds, the sprits of the seasons also appear in seasons different from their 'assignments.'" Lily quoted 'assignments,' as the seasonal duties weren't actually assigned as such.
"Aren't you supposed to know this, Herald?"
"Why should I? I just announce spring. I don' bring it."
76. What Winter Left Way Behind
"Letty Whiterock is the Spirit of Winter. As a seasonal spirit, that means she knows which season it is at all times, or rather she should," Letty began, holding up a lecturing finger. "Every seasonal spirit I've encountered, sane or insane, believes exactly one season is the correct one. The same will be true of Letty."
"Are you sure?"
Lily shrugged again. "I don't know how to explain it, but it is one thing I am dead certain of. No doubt in my mind at all," she assured.
"More implanted memories."
"I guess," Lily agreed. "Anyway, it's important that a seasonal spirit knows which season it is. If they don't have their seasons right, we'll have a struggle between seasons, as two seasonal spirits try to assert their season, or a season without a season, as it were."
"I suppose that's bad."
"Very! You get crop failures and poor harvests and humans get really really rowdy after that."
Remilia drew back, looking strangely at the spring fairy. "And how would you know about crop failures?" she asked, genuinely curious.
"Hey, I've been around long enough to know that humans need food and farming is how they get a good chunk of it. I also know that spring is the time when they start planting." Lily's eyes were narrowed at the vampire. "'M not completely dumb, ya know," she insisted.
Remilia huffed, "I never said that." The vampire turned to Letty, who was standing passively where they found her, not making a sound. "You've been pretty silent, Letty, so answer me this question: Yes or no, are you either a knight who will answer 'no' to this question, or a knave who will answer 'yes'?" Remilia asked at last.
Letty gave her answer, and it was one of 'yes' or 'no.' Her answer made Remilia grimace with consternation. "Ladies, if this Incident continues, we may be in for some problems," Remilia said aloud, an image of an asesonal winter firmly in her mind.
Solution: By the very fact that she answered the question, either 'yes' or 'no,' Letty has proven that she is insane!
If Letty is a knight and answers 'no,' then she is a knight who answered 'no' to Remilia's question. This makes the question's proper answer 'yes,' which means that Letty's answer is wrong. If Letty instead answers 'yes,' then she is not a knight who answered 'no' to the question, neither is she a knave who answered 'yes' to the question (since she isn't a knave at all), and therefore Letty's answer is still wrong. There is no way that Letty can correctly answer the question, 'yes' or 'no,' if she is a knight, so since she does answer, she must have answered wrongly and must be insane.
If Letty is a knave and answers 'yes,' then she is a knave who answered 'yes' to the question. This makes the question's correct answer 'yes,' which means that her answer is correct. If Letty instead answers 'no,' then she is neither a knight who answered 'no' (since she is not a knight), nor a knave who answered 'yes,' and therefore the correct answer is 'no,' which is her answer. There is no way that Letty can answer incorrectly, 'yes' or 'no,' if she is a knave, so since she does answer, she must have answered correctly and again must be insane.
Thus, knight or knave, if Letty answers the question at all with a 'yes' or a 'no,' she is insane! (The only way a sane inhabitant can escape this if she refuses to answer!)
—/—
The Ghostly Umbrella and the Two-Eyed Satori
"So, we've discovered that Letty is a nutter, and we have in our possession a pair of keys of untold function, and a pair of sane and knightish fairies," summarized Remilia as the group made their way deeper into the forest. "I wonder what these keys are for."
Lily shrugged her shoulders, and Honey simply remained silent on this point.
"All we can do is demand answers now," stated Sakuya, falsely.
"BOO!" shouted the humanoid half of the youkai known as Kogasa Tatara, the Cheery Forgotten Umbrella.
Neither Remilia nor Sakuya jumped in surprise, but Kogasa found that Sakuya's reflexes were well honed, as a knife embedded in her forehead.
"Ow," Kogasa said, pained as she extracted the knife. Whether this was because she was a youkai and was just that tough, or because the humanoid body wasn't her actual self, was anyone's guess.
"Hello!" chirped Koishi Komeiji, the Closed Eyes of Love, unconsciously as she simply appeared out of nowhere.
"Ah, Kogasa Tatara and Koishi Komeiji," said Honey Humblebee aloud. "I've been expecting you, as we have business to square away with you. Or, more accurately, Lady Remilia Scarlet has business to square away with you."
"Usual rules?" asked Remilia.
"Usual rules," confirmed Honey.
76. First Round of Statements
"I suppose I'll start with this well-worn instruction: please tell me about yourselves or your companion," Remilia began.
Kogasa smiled, saying, "I believe Koishi is sane."
"Kogasa is a knight," added Koishi.
Honey added in her own hint, "These two are opinionated on their types and sanities."
Remilia thought about it for a minute, then announced, "This isn't nearly enough information. I can only figure out Koishi's type!"
Solution: All we can tell at the moment is that Koishi is a knight.
Suppose Kogasa is telling the truth, and she really believes that Koishi is sane. Then Kogasa is either a sane knight or an insane knave. If she is the former, then since Kogasa is sane, it is true that Koishi is sane. This also makes Koishi's statement true, since Kogasa really is a knight, so Koishi must be a sane knight. If Kogasa is the latter, then her belief that Koishi is sane implies that it is false, so Koishi is actually insane. This also makes Koishi's statement false, so she must be an insane knight. Thus, if Kogasa really believes Koishi is sane, then Koishi is a knight, and has sanity equal to Kogasa.
Suppose instead that Kogasa is not telling the truth, and she doesn't really believe that Koishi is sane. Then Kogasa is either an insane knight or a sane knave. In addition, Kogasa is opinionated on Koishi's sanity, so she either believes that Koishi is sane, or that Koishi is insane (not sane). She does not believe that Koishi is sane, she must believe that Koishi is insane. If Kogasa is an insane knight, then her belief that Koishi is insane is not true, so Koishi is sane, and because Kogasa is indeed a knight, Koishi tells the truth, so she is a sane knight. If Kogasa is a sane knave, then Koishi is indeed insane. Also, Koishi has told an untruth about Kogasa's type, so Koishi must be an insane knight. Thus, if Kogasa doesn't really believe Koishi is sane, then Koishi is a knight, and has opposite sanity to Kogasa.
In all cases, Koishi is a knight, and Koishi and Kogasa match in sanity if and only if Kogasa really believes Koishi is sane.
77. One Question
Having only figured out one thing about one of the two, Remilia decided on her follow-up. "Is Kogasa sane?" the loli vampire asked the satori.
"I believe so," came her answer.
"Well, I don't!" countered Kogasa.
Remilia smiled. She now knew exactly what each of them were.
Solution: Koishi claims, not that Kogasa is sane, but that she believes Kogasa is sane. Because she is a knight, she believes she believes Kogasa is sane. If Koishi herself is sane, then it is true that she believes that Kogasa is sane, and hence, it must be true that Kogasa is sane. If Koishi is insane, then it is not true that she believes Kogasa is sane. But she is opinionated on Kogasa's sanity, so she believes Kogasa is not sane, but since Koishi's insane, this belief is false, so again Kogasa is sane.
Kogasa denies she believes she is sane. But she is sane, so she cannot believe she is not sane. Since she is opinionated on her sanity, she therefore believes she is not not-sane, or sane. She therefore does not speak the truth, but is nevertheless sane by Koishi's answer. Therefore, she is a sane knave and she lies.
Because she lies, she doesn't really believe Koishi is sane (see previous solution), and we have already deduced that if that is the case, then Koishi is of opposite sanity to Kogasa, and hence insane.
—/—
The Kirisame Magic Shoppe
Remilia marked off a couple of ticks on her To Do list, now that she has found the complete types of both Kogasa and Koishi. "Okay, that leaves the Broom-Riding Thieving Intruder and the Seven-Colored Puppeteer," she stated.
"Marisa's place is not around that bend," Sakuya pointed out, breaking silence once more. Remilia noted that her maid had become a woman of few words since the incident began, owing to her now knavish nature. Sure, the maid had some fun early on, but seemed to have tired of that quickly and fell mute. Remilia felt sort of sorry for her, but there was nothing to be done but to resolve the incident, and that meant doing things Yukari's way for now.
Marisa's place was sited in the clearing that they just emerged from, and 'place' was about as truthful a way to characterize the domicile without getting insulting. As always, it was characterized by its poor structure. Looking at it, it was easy to believe that Marisa would not emerge for days from it, because it was likely that the mushrooms that she was rumored to eat regularly would grow on the inside of the structure, using the very members that the structure was built from as nourishment.
Remilia was in no mood to laugh at the hilarious structure, even though the urge was definitely there. She instead strided up to it, Sakuya hurrying after her to hold the parasol over her head. With one pale hand, she knocked smartly on the door, which seemed more solid and sound than it appeared.
The door opened, revealing the blonde-framed face of Marisa Kirisame. She was also framed by a few lengths of bandages around various limbs and her head, covering one eye, and her left arm hung limply in a sling. In short, in comparable condition to the structure she lived in. "Yo, Remilia! What can I do you for?" she greeted the vampire and maid with a wide, but pained grin on her face.
78. The Ordinary Magician and the Fairy
Remilia cut right to the chase. "I want answers to two questions: First, are you a knight? And second, are you sane?"
Marisa stared at Remilia, flatly and without answering. Remilia could tell that she had been asked these two questions before. Was she quizzing herself whether or not to answer the questions? Then Marisa grinned, seeing Honey Humblebee. "She knows those answers," she replied, pointing at the companion fairy.
"C'mon! Just tell me, dangit!" Remilia snapped.
"Do I look like the sort of person who would wantonly give out easy hints?" she asked.
Remilia looked into the black-white's eyes, crinkled almost shut by the wide grin just below it. "No," she answered simply. She turned to Honey Humblebee, "Well? Do you know whether or not Marisa's a knight or knave, sane or insane?"
"I'll only say that Marisa is not an insane knave. You also have quite enough information to know the type and sanity of Marisa," Honey replied with a strange smile on her face.
Remilia opened her mouth to protest that it wouldn't be nearly enough, but clamped her mouth shut. "I am faced with two incredible facts, and I mean 'incredible' in both senses of the word," Remilia remarked after thinking about the matter carefully.
Solution: Marisa is a sane knight, and Honey Humblebee is omniscient!
Someone is omniscient if they know all propositions to be true or false. Someone knows a proposition is true or false if that person believes that proposition if and only if that proposition is true.
Take any proposition, S. Since Honey is sane, if she believes S, then S is true. Conversely, if S is false, she does not believe S. Furthermore, she is opinionated, so for every S, she believes exactly one of S or not-S. If S is true, then not-S is false. Therefore, Honey, being sane, does not believe not-S. But if she does not believe not-S, then she does believe not not-S (because she is opinionated), and therefore believes S. Thus, if S is true, Honey believes S.
Honey therefore believes S if and only if S is true. Since this applies to any S, for all propositions, Honey believes that proposition if and only if it is true, and any being that fulfills this criterion is omniscient.
Marisa is not an insane knave, because Honey said so, and she is omniscient and a knight. However, Marisa did say that Honey knew the answers to Remilia's questions. Honey is omniscient and knows everything, including the answers to Remilia's questions, so Marisa's statement is true. Since she is not an insane knave, yet she utters true statements, she must be a sane knight.
—/—
Alice's Army of Fairy Dolls
"So, Alice's Army of Dolls are on the loose, you say?" began Remilia.
"Indeed," replied Marisa, "They're not exactly causing trouble, but they are very cleverly and aggressively defending their master's house."
Marisa treated Remilia and her entourage to mushroom tea and mushroom soup. It was a lot better than it sounded, Remilia had to grudgingly admit. Maybe it had some bizarre hallucinogen in it that bamboozled her into believing it to be good, which would go a long way towards explaining why Marisa ate them. She took another sip of the slightly thick tea.
"And what have you found out about them?" Remilia asked.
Marisa ticked off the points as she went through them. "First, that Alice's dolls can change their lying and truth-telling behavior and their beliefs, depending on the day and the week. The dolls come in four types, which I've labeled K, N, 1 and 2. You'll understand why later. They also come in four classes, labeled by yours truly, S, I, A, and B. Again, you'll understand why later." Remilia frowned at the glossing over of those details, but said nothing. After all, Marisa said she would understand later, and Marisa was a sane knight. Marisa continued, "Each doll, whatever other characteristics they have, belong to exactly one of these classes and exactly one of these types."
"So, a doll can be of designation A2, or SK, or IN?" Remilia asked. Marisa nodded in the affirmative, and Remilia smiled a bit, satisfied with the explanation. "But what do the designations mean?"
"That's what I'm about to explain next. At any one time, each doll is in one of three states, which I call 'stages': 1, 2, or 3. In some of these stages, which I call 'truthful stages,' the doll is in what I call a 'truthful mood,' where she never makes a statement or answers a question contrary to her knowledge and beliefs. In the remainder of these stages, she always makes statements and answers questions contrary to her knowledge and beliefs - she's in a 'deceitful mood,' and the stage called a 'deceitful stage.' A K-type doll is never in a deceitful mood, only in a truthful mood, while an N-type doll is never in a truthful mood, only in a deceitful mood."
"I get it. 'K' and 'N' stand for 'knight' and 'knave,'" Remilia summarized.
Marisa touched her nose as she pointed to her, indicating that she had caught on. "Type 1 dolls, however, have stage 1 as their deceitful stage, while the remaining two stages are truthful, and dolls of type 2 have stage 2 as their deceitful stage, with the remaining two stages truthful."
"I guess that the classes refer to the behavior of sanity in the dolls."
"Correct," replied Marisa. "At any one time, a doll is in one of three other states, which I call 'phases': A, B, and C. Some will be 'lucid phases,' where all the doll's beliefs are correct, and others will be 'deranged phases,' where all the doll's beliefs are incorrect."
Remilia picked up the obvious from there. "If I had to guess, class S dolls are always lucid, class I dolls are always deranged, class A dolls are deranged only in phase A and lucid otherwise, and class B dolls are deranged only in phase B and lucid otherwise."
"Give the vampire an explodin' cigar!" shouted the Black-White jovially. She sobered quickly, adding, "Two more rules: First, a doll believes S in her deranged phase if and only if she believes not-S in her lucid phase."
Remilia interrupted Marisa at that point before she could continue to the next rule. "Then changing phases preserves opinionation. If a doll is opinionated on some S while in some phase, then they are opinionated on that same S in any other phase," concluded the vampire.
"Right. Second rule, each doll is completely opinionated on various attributes of both themselves and their sister, including their types and classes, their states and phases both past and current, and whether they are currently reliable or unreliable."
"'Completely opinionated'? 'Reliable'? 'Unreliable'?" Remilia quizzed. More new terms.
"'Reliable' and 'unreliable' are my terms for a doll that currently makes true statements and currently makes false statements respectively, whether they are actually truthful, lucid, or anything else — it's helpful in keeping track of the mental gymnastics," answered the magician.
Remilia carefully sipped up a spoonful of mushroom soup, noting its crude but hearty taste as she contemplated the new term. "Actually, that wouldn't be a bad general term for other kinds of beings, too," she said aloud, noting that several internal reasonings could have been made shorter by the use of the terminology. A second later, she cursed, "Crap. That probably means it'll go into general use, now that I've said it."
Marisa didn't press that observation. "'Completely opinionated on S' is a bit more difficult to define, but its a statement about what I call belief chains based on that statement S. A direct belief in S and a direct belief in not-S are both belief chains based on S. But if we have an X that we know to be a belief chain based on S, then a direct belief in X or belief in non-X is also a belief chain based on S. If you are completely opinionated on some S, then you are opinionated on all belief chains based on S."
Remilia thought about that for a moment. "So if I am completely opinionated on my reliability, then I am opinionated on my belief that I believe that I am not reliable, and also on any chain of belief and negation as long as it has something to do with my reliability."
"Yep."
"Leaving that aside, that these types and classes apply differently to differing stages and phases strongly suggest that a doll will change both at some time or another," Remilia stated aloud, having a strong suspicion where this was going.
Marisa did not disappoint. "Yep. A doll's stage goes from stage 1 to stage 2 to stage 3 then back to 1, changing stage every twenty-four hours at midnight. A doll's phase goes from phase A to phase B to phase C and then back to A, changing phase every week when the sun rises on Sunday. Together, the combination of a doll's stage and her phase is her state: B2 is a state, as is A1 and C3."
Remilia narrowed her eyes. Something about Marisa's testimony did not add up. "Wait a second," she said in confusion, "the current incident hasn't been going on for weeks and weeks."
"Maybe not the incident proper, but Alice's current project started months before. It was quite innocuous and actually kind a' charming, but then this knight/knave, sane/insane thing hit us and it was clear that Alice's dolls were a prototype. Wasn't so charming after that."
"I see."
"Back to the dolls." said Marisa, a little loudly, to steer the discussion back on track. "You know how Alice makes dolls according to standardized doll lines?" The witch noted Remilia's nod. "Each line has a sister line, and each doll in a line is paired with a doll from the sister line. The dolls are made two at a time, one from each sister line, and form a 'matched pair,' and the dolls in a matched pair are called 'sisters.' The dolls of a matched pair will always be in the same stage and same phase as each other, but this is not necessarily true for a doll of the same or their sister line but not from the same matched pair.
"In order to pass these dolls, you haveta solve a specific puzzle they will pose to you. If they do not pose a challenge to you immediately, then you get to ask questions until they do. The only exception are Shanghai and Hourai. Also, when they are encountered outside of Alice's house, they are either single, or paired with their sister."
Marisa and Remilia talked about specific lines of dolls at that point, which Remilia filed away in her mind for later. They will remain unimportant until the correct time.
Marisa wrapped up the explanation, "As you get deeper into the forest, the puzzles get harder. If you don't solve each dolls' puzzle, she won't let you pass."
"Sounds like a lot of trouble. I think this is a time to break out the danmaku and take care of them the old fashioned way."
"Don't," the ordinary magician warned. "If you try to get past them without answering their challenges, and especially if you use danmaku on them, all of Alice's dolls will converge on you... and when I say 'all of Alice's dolls,' I do mean all of them... and they will wreck your shit!"
Remilia regarded Marisa's state of health, as did Sakuya. "Is that how you ended up in such a sorry state, Marisa Kirisame?" asked the maid, who was obviously expecting an answer of 'yes.'
"Well... yeah."
"Huh, if they waxed a Forth Stage like yourself, they'll certainly be trouble for us," Remilia snarked lightly, then taking a sip of her mushroom tea.
"Uh, Remi. I'm actually Extra Stage certified, y' know."
Mushroom tea emerged in a fine mist as Remilia spat it out on reflex. Coughing a few times to clear the remainder, Remilia slammed her hands onto the table as she leaned over it with an intense look of disbelief on her face. "You're an Extra Stage Boss!" she shouted.
"Yeah. Had the cert' for years. Alice's Extra Stage, too, an' I've had to face it. Reimu holds Sub-Phantasm certification," said the blonde human magician calmly.
"But... but... you were Forth Stage during the Stolen Moon Disaster!" Remilia sputtered.
"Well, you know ya don't have to play at your rated stage all the time!" Marisa said, a smile finally appearing on her face in enjoyment of seeing Remilia's charisma breaking before her eyes. "Yuyuko, a Stage Six at least, played at Stage One during that Mausoleum incident just a few weeks ago."
Whatever charisma Remilia had lay in shambles at her feet. Marisa, a mere human witch and a thief to boot, was rated at a higher stage than her, the Queen of the Night! It was quite shocking to the vampire. After a moment of barely restrained weeping, she wheeled about and pointed at Sakuya, who retained her composure. "SAKUYA! MAKE A NOTE!" she shrieked, "As soon as this incident is over, I'm training for Extra Stage certification!"
"It will not be done, m'lady," Sakuya said.
Marisa stared at Sakuya with narrowed, suspicious eyes. A crafty smile creeped to her lips. "Sakuya, are you sane?" she asked, out of the blue.
"No," Sakuya answered, then immediately winced in realization.
"Ah. Yer a knave," Marisa said matter-of-factly, grinning at Sakuya's unwitting reveal. "My condolences."
79. The Locked Box of Marisa Kirisame
Marisa rose from her place and weaved around the piles of junk that littered her house outside this small little 'living' area. She took up a small box with a brass-detailed keyhole on its front face. "One more thing: this strong-box was left on my doorstep this morning, with instructions that I was to give it to you," she said as she set it down before Remilia. "The note said it was a problem about keys."
Remilia took it and applied a gentle pull to its top. It rattled, but refused to open.
Sakuya glanced at the locket and hazarded a guess. "M'lady, it is not probable at all that the keys you have received from the fairies will be useful for opening this locked box."
"Be careful, Lady Scarlet. You have to use a specific key, and turn that correct key in a specific direction — clockwise or counterclockwise, or the lock will permanently jam," warned Honey Humblebee, the omniscient fairy.
Remilia huffed, merely laying both keys out on the table with a clatter: ward-lock keys cast in gold or brass, one with a diamond shaped bow, the other with a heart shaped bow. But which one should she use, and which way should she turn?
"You'll notice that bolted to the top of the box are four plates of differing metals —gold, silver, copper, and iron— and each engraved with the chemical symbol of their elements and three statements," Honey continued. Remilia did not need to look at it to see that it was so, Honey being a sane knight, but she looked anyway. As expected, it was so. "Each plate has been inscribed in its entirety by either the left or right hand of a single skilled kappa, Eta the Kappa, who is a hermit living in this forest. That is to say, for example, if the kappa's right hand has engraved a single inscription in the copper plate, then the kappa's left hand has not engraved any inscription in the copper plate."
"Is this kappa sane or insane?" asked Remilia.
"That's for you to decide, along with whether Eta is right- or left-handed, or indeed a he or a she. I assure you that Eta is one or the other of those," answered the omniscient fairy.
Remilia looked over the inscriptions at this point.
On the gold plate was inscribed the following:
~Au~
(1) Gretta Julibee is an outsider fairy and claimed to be either sane or insane, and also claimed that either the diamond-bowed key or the heart-bowed key is the correct key for this lock.
(2) No more than two of these plates have been inscribed by my right hand.
(3) I believe that the diamond-bowed key is the correct key only if the correct key must be turned counterclockwise to open the lock.
For the silver plate:
~Ag~
(1) I am female.
(2) I am right-handed, and I am insane.
(3) Gretta Julibee claimed that the diamond-bowed key is the correct key for this lock.
For the copper plate:
~Cu~
(1) Actually, I am male, or I am right-handed.
(2) Inserting the heart-bowed key and turning it clockwise will jam the lock.
(3) This plate was inscribed with my right hand.
For the iron plate:
~Fe~
(1) If I like green tea, then I am insane.
(2) Gretta Julibee claimed to be sane.
(3) You will arrive at Alice's home and when you get to Alice's home, cake will be served to you, in your honor.
Remilia clucked her tongue. "Typical. At least the inscriptions bode well for the kappa race," she said as she chose a key and twisted it in the lock.
Solution: The diamond-bowed key is the correct key for the lock, and it must be turned clockwise.
First, let (X N) denote the N'th inscription on the plate of elemental symbol X, so the third inscription on the gold plate is (Au 3).
(Ag 1) and (Cu 1) cannot both be false, because if they were, then Eta is neither male nor female, and Honey said Eta was one or the other. The only way both inscriptions (Ag 1) and (Cu 1) can be true is if Eta really is female and right-handed. By (Ag 2) Eta must also be insane. This means that Eta's left hand is reliable, and inscribed both plates, while Eta's other hand is not. But (Cu 3) claims that it was inscribed by Eta's right hand, which cannot be because Eta's right hand would be unreliable and inscribe only false inscriptions. The assumption that (Ag 1) and (Cu 1) are both true leads to a contradiction. One must be false.
Suppose the silver plate has true inscriptions. Then the copper plate has false inscriptions. By (Cu 1), Eta is neither male nor right-handed, and thus must be left-handed. But this contradicts (Ag 2), which asserts (if it is true) that Eta is indeed right-handed. Therefore, the silver plate is not true, and from (Cu 1), we can infer that the inscriptions on it are true (because Eta is not female, but male — contrary to the false inscription (Ag 1)). We can also infer from (Cu 3) that the copper plate has indeed been inscribed by his right hand, and the silver plate by his left.
(Ag 2) is not true, which means that Eta is either left-handed or sane. Suppose he's left-handed. Then by (Cu 3) Eta is insane, because a left-handed person can only write truth with their right hand if he is insane. Suppose however that Eta is right-handed. Then by (Ag 2), he is sane. Therefore, Eta is sane if and only if he is right-handed.
We already know the silver plate has only false inscriptions, so if the gold plate bears false inscriptions too, then there are only two plates left that could possibly bear true inscriptions. But this is what inscription (Au 2) implies: that at most two of the plates bear true inscriptions. Therefore, if the gold plate bears only false inscriptions, then one of the inscriptions the gold plate bears is true — a contradiction. Therefore, the gold plate must bear only true inscriptions. Since it is true that at most two of the plates bear only true inscriptions, and the gold and copper plates are those two plates, the iron plate must bear only false inscriptions.
All inscriptions on the iron plate are false, so the conditional statement (Fe 1) is false too. The only way a conditional can be false is if the antecedent is true, and the consequent is false. Therefore, the consequent that Eta is insane is false. Eta is sane. Also, the antecedent is true, so Eta likes green tea... though that's not important for this problem. Eta is therefore a sane male kappa, which bodes well for the kappa race.
The copper plate has true inscriptions, therefore by (Cu 2) inserting the heart-bowed key and turning the it clockwise really will jam the lock. Therefore, it cannot be the case that the heart-bowed key is the correct key and the correct key must be turned clockwise are both true. Either the key must be turned counterclockwise, or the heart-bowed key is the incorrect key.
The gold plate has only true inscriptions, so by (Au 3) Eta really does believe that the diamond-bowed key is the correct key only if the correct key must be turned counterclockwise. We've already established that Eta is sane, so this belief is correct. If the correct key must be turned clockwise to open the lock, then the diamond-bowed key cannot be the correct key; the heart-bowed key must be the correct key. But this cannot be, for by (Cu 2) the heart-bowed key jams the lock if turned clockwise. Therefore, the correct key must be turned counterclockwise.
Inscription (Au 1) tells us that Gretta Julibee is an outsider fairy, claimed to be either sane or insane, and claimed that either the diamond-bowed or the heart-bowed key was the correct key to the lock. The inscription is on the gold plate, which has no false inscriptions. Therefore, Gretta is indeed an outsider fairy, and as such she is sane. She has also claimed to be either sane or insane, and separately named one of our two keys as the correct key. However, inscription (Fe 2) claims that Gretta has claimed to be sane, but being on the iron plate, it is false. Therefore, Gretta has claimed to be insane, which makes her a knave.
Inscription (Ag 3) claims that Gretta claimed that the heart-bowed key is the correct key to the lock, but since the inscription is on the silver plate, it is false: by elimination with (Au 1), Gretta actually claimed that the heart-bowed key is the correct key to this lock. But Gretta is a sane knave, therefore, the heart-bowed key is not the correct key; the diamond-bowed key is, and it must be turned counterclockwise.
Finally, (Fe 3) claims that Remilia will arrive at Alice's home, whereupon they will receive cake, but all inscriptions on the iron plates are false. Therefore, either Remilia will not make it to Alice's house, or if she does arrive, there will be no cake served.
Yes, it does mean the cake is a lie. I'm very, very sorry.
—/—
The lock released obediently, as Remilia expected, and the vampire opened —in a somehow smug— way the wooden box that contained the secret. The interior of the box was lined with green felt, with two cavities that exactly matched two further keys, again cast in gold or brass, one with a spade shaped bow, the other with a clubs shaped bow — like card suits. Both keys had a black gem embedded in their bows, perhaps an onyx or a jet. Remilia frowned at these contents.
"Oh, bother. More keys," she murmured. "At least with any luck, we won't need to bother with the two old ones." She unrolled and read the note that came along with the keys. "Curse that troll," growled the loli vampire as she finished the note, "It says the keys we already have 'might be useful' in future. So much for luck. Come, Sakuya, Lily, and Honey."
Marisa hobbled up to her doorway, and leaning on the door frame, called out, "Don't forget what I tol' ya about them dolls!"
"Yeah, yeah."
—/—
Interlude: Deeper in the Forest of Insanity
"Come, Remilia Scarlet... Sakuya Izayoi..." hissed a female figure cast in shadow, "Come into the glorious bliss that is insanity!" She let out a long, loud peal of cackling.
On the other side of the room, a doll dressed in red shook a thin, wide metal sheet, making a crashing sound that rolled across the room. Two dolls dressed in blue and wearing safety goggles attended a flask where a measure of magnesium burned brightly, as a third rapidly opened and shut a blind that selectively permitted and blocked that magnesium-originated illumination in a flashing staccato upon the mysterious figure.
(To be continued... unless I'm INSANE!)
Notes: I split this chapter up, owing to the ever burgeoning length and the fact that the last part of it is having gelling problems. The conclusion to Remilia's journey through the Forest of Magic will be soon.