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, accompanied by Lily White and Honey Humblebee, are making their way towards the abode of Alice Margatroid to assay her type and sanity, as well as find out what she was up to. The abode is guarded by rings of dolls, which Marisa warned not to challenge with danmaku (citing her copious injuries).

The dolls had the peculiar property of shifting their lying habits depending on which day it was, and their sanities depending on the week. Each doll cycled through three stages, 1, 2, and 3, progressing through these stages one per day. Each doll also cycled through three phases, A, B, and C, progressing through these phases one per week. Each doll's stage and phase is synchronized with a counterpart from a sister line, the doll's 'sister.' When Alice's dolls travel, they are always singly or paired with their sister.

Each doll had a type that dictated how the doll will behave during that day. A type K doll (K for 'knight') is always 'truthful;' they speak according to their beliefs. In contrast, a type N doll (N for 'knave') is always 'deceitful;' they speak contrary to their beliefs. If the doll's type was 1, the doll is deceitful for as long as they remain in stage 1, and truthful otherwise; similarly, type 2 dolls are deceitful in stage 2, and truthful otherwise.

Each doll also had a class that dictated their sanity for that week. A class S doll (S for 'sane') is always 'lucid;' every belief they have is true. A class I doll (I for 'insane') is always 'deranged;' every one of their beliefs is wrong. Class A dolls are deranged in phase A and lucid otherwise, and similarly class B dolls are deranged in phase B and lucid otherwise.

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, thus they are opinionated on all belief chains based on those attributes. Furthermore, a doll that is opinionated about some subject while lucid will remain opinionated on that subject while deranged.

Marisa had also given her particular information about each line of dolls. The party will use that information to navigate each ring of dolls as they come to them. While Remilia did not know what to expect at Alice's house, she did know that there will be no cake served once she arrived.

Knights and Knaves:

A Logical Adventure of Gensokyo

by Wyrm

Chapter 7:

In Which Remilia Scarlet Confronts the Seven Colored Magician

Dolls from the Siberian Wastes ~ Tibet and Russia

"Tibet and Russia are the first sister lines. Tibet line dolls dress in a traditional tibetan style, while Russia line dolls dress —surprise, surprise— in traditional russian dress. All Tibet dolls have a designation of S1, and all Russia dolls, S2."

These were the words of Marisa Kirisame concerned with the Tibet and Russia line of Alice's dolls. "This is the first time we will be encountering a being that actually changes the way they lie or tell truth," Remilia said aloud, even though she knew Sakuya had probably deduced this.

A sign was posted beside the path, loudly advertising that they were entering the domain of Alice's dolls. "Come on, my friends," Remilia said as she and Sakuya strolled forward, followed by the hovering Lily and Honey, entering hostile territory.

80a. Together with Tibet and Russia

It didn't take long for them to encounter a pair of these dolls, one Tibet doll and one Russia doll. According to Marisa's information, this would be a matched pair.

Remilia recalled that Tibet and Russia were always lucid, and together in one of three stages, 1, 2, or 3, that would progress by one stage every day. Tibet (being type 1) would lie (be deceitful) in stage 1, and tell the truth on other days. Russia (being type 2) would lie in stage 2, and tell the truth at other times.

Without any prompting, Tibet and Russia made their statements.

"Yesterday I was in a deceitful mood," said Tibet.

"Yesterday I was in a deceitful mood too," added Russia.

Tibet then issued their challenge: "Will either of us be deceitful tomorrow, and if so, who?"

Remilia smiled. It wasn't that difficult a challenge, but then, it was only the first ring.

Solution: If Tibet is reliable today, she is telling the truth and was deceitful yesterday. The only stage where this is true is stage 2. If Tibet is unreliable today, she was not deceitful yesterday. The only stage where this is true is stage 1. So the dolls' current stage is 1 or 2.

If Russia is reliable today, then she was deceitful yesterday. The only stage where this is true is stage 3. If Russia is unreliable today, then she was actually truthful yesterday. The only stage where this is true is stage 2. So the current stage is 2 or 3.

Thus, the current stage can only be 2 for this doll pair. Tomorrow, they will be in stage 3, so neither doll will be deceitful tomorrow.

80b. A Lone Tibetan Doll

A while later, the group encountered another Tibet doll. She was alone; if her sister was about, she was not near.

"I was in a deceitful mood yesterday," she began, and added, "I will be in a deceitful mood again tomorrow." She then issued her challenge: "What stage am I currently in?"

Remilia solved this one easily, too.

Solution: Her statements are either both true or both false. If she's telling the truth both times, then she is in a deceitful stage two days in a three day cycle, which a doll of type 1 does not have. Therefore, she is deceitful today, and since she is type 1, she must be in stage 1.

80c. Solitary Russian Doll

A few minutes later, they encountered another of the dolls. This time, it was a Russia doll. There was no guarantee that this doll was the sister to the Tibet doll they encountered previously.

"If I was in a deceitful mood yesterday, then I will be in a deceitful mood again tomorrow. What stage am I in?" it challenged.

Remilia did not find it that much of a challenge.

Solution: If Russia was in a deceitful mood yesterday, then her stage has advanced since yesterday and she is in a truthful mood today. This means that the implication is true, and the antecedent is true, which implies the consequent is true: that she will be in a deceitful mood tomorrow. But tomorrow the doll's stage will have advanced twice from her deceitful stage, and her next deceitful mood is on the day after that! Therefore, yesterday she was not in a deceitful mood.

This makes the antecedent false, and as such the entire implication is true (an implication with a false antecedent is always true). Therefore, today she is still not in a deceitful mood, and because yesterday she was also not in a deceitful mood, then tomorrow she will be in a deceitful mood (as her deceitful moods come in a three day cycle). Since Russia is a type 2 design, tomorrow she will be in stage 2. Therefore, today she is in stage 1.

—/—

Italian Dolls from the Cities ~ Napels and Turin

"Napels and Turin are the next sister lines. They dress identically in sea green Florentine dresses, and the dolls themselves have black hair both. They are indistinguishable, so don't bother identifying them visually. Napels has the designation S1, and Turin, S2. They're usually alone."

Marisa's words meant that Napels and Turin basically followed the same rules as Tibet and Russia, only this time they were indistinguishable from each other.

81a. An Italian Doll by Herself

The first time they encountered one of these dolls, she was alone. Remembering Marisa's warning, she could have been a Napels or Turin doll.

"I am in a deceitful mood today, or I am Napels," the doll announced.

Remilia thought about it. She could deduce one thing about the doll already, one vital datum required for unharassed passage, but she could not figure out the other. "Is your sister deceitful today?" she asked.

The doll answered, "Yes." Then the doll made her challenge: "Will either of us be deceitful tomorrow?"

Solution: If the doll is in a deceitful mood today, then her statement is false, and she is neither Napels nor in a deceitful mood. But this is a contradiction, so the doll cannot be in a deceitful mood; she must be in a truthful mood. And since she isn't in a deceitful mood, she must be Napels.

Napels is a type 1 doll, and her sister is type 2. Since she is in a truthful mood today, her answer to Remilia's question of whether her sister is in a deceitful mood is the truth: her sister Turin is in a deceitful mood, which means the dolls are in stage 2. Tomorrow, they will be in stage 3, and thus neither will be deceitful.

81b. Another Italian Doll by Herself

It was not long before they encountered another of these dolls. She looked identical to the last doll, Italian dress and all. "I am in a deceitful mood today, and I am Turin," she announced. "What stage am I?"

"At least this one is fairly easy," remarked Remilia before answering the doll's challenge.

Solution: If the doll is in a truthful mood, then she is certainly not in a deceitful mood, which means that her statement is a lie. But this is a contradiction, so she must be in a deceitful mood today. Since she is in a deceitful mood, she cannot be Turin (otherwise her statement is true), so she must be Napels. Since Napels is a type 1 doll, and this Napels doll is in a deceitful mood today, she must be in stage 1.

—/—

Draw Me Like One of Your French Dolls ~ Nancy and Nice

"Nancy and Nice are yet another pair of sister lines. They are virtually indistinguishable, save for their dresses, which are colored blue and green, both in french fashion. However, they often exchange garbs, so don't depend on them. Also, while I've figured out that one doll is designated S1, and the other S2, I have no clue which doll has which designation."

Remilia had remarked at that point that the dolls were named after cities or countries, either in Europe, or in Asia. Marisa expressed the fact that he didn't particularly care about that. As to the dolls themselves, they were like Napels and Turin, except that the designations were unknown.

82a. The First Conversation with Nancy and Nice

Two dolls in matching green and blue french dress rose to block the group, Nancy and Nice. Remilia decided to designate the doll dressed in blue, 'Doll B,' and the doll dressed in green, 'Doll G.'

Doll B said, "If I'm Nancy, then she's Nice."

In her turn, Doll G said, "If she's Nice, then I'm Nancy."

Doll B issued the challenge: "What is our current stage?"

Although Remilia could not tell which doll was Nancy and which was Nice, she was able to name their stage, and so answer their spoken challenge.

Solution: If Doll B is Nancy, then Doll G is Nice; and if Doll B is Nice, then Doll G is Nancy. This is obvious from the fact that dolls are paired up. However, this is precisely what the dolls assert, so the both of the dolls are in a truthful mood. The only stage where both dolls are truthful is stage 3. It is impossible to tell who was who or their individual designations, however.

82b. All One Ever Wanted to Know about Nancy and Nice

Remilia spotted the second pair of Nancy and Nice dolls before they could drop off their perch on a high branch to confront them. "Hello," greeted the immature vampire. The two dolls nodded in greeting, then descended to confront the four travelers. In her mind, Remilia again labeled the green-wearing doll 'Doll G' and the blue-wearing doll 'Doll B.'

Doll B began, "Today we are not in stage 3."

Doll G followed, "In fact, today we are in stage 2."

Doll B gave the final statement. "Tomorrow, Nancy will be in a deceitful mood."

Doll G issued the challenge: "Figure out (1) the lines of each of us, (2) what stage we are in, and (3) our types."

Remilia grinned. Those three statements were quite sufficient to answer the green doll's challenge.

Solution: It is impossible for the two dolls to be in stage 3, because if they were, Doll B could not lie and say that they were not in stage 3. Therefore, they are in stage 1 or 2. Doll B is therefore telling the truth — they are indeed not in stage 3. Thus, Doll G is deceitful, so they are not in stage 2. This leaves them in stage 1, Doll B is type 2 and Doll G is type 1.

Since Doll B tells the truth today, Nancy will indeed be in a deceitful mood tomorrow. By tomorrow, however, the dolls will have transitioned to stage 2, which means Doll B will be in a deceitful mood tomorrow, while Doll G will be back in a truthful mood. Therefore Doll B is Nancy, who is an S2 doll and Doll G is Nice, an S1 doll.

—/—

Country Dolls of the West ~ France and Holland

"Then we have the sister lines, France and Holland. These two lines are easily distinguishable, with France wearing a flowing green frilly french dress, while Holland wears a black netherlands dress with a white apron. Of course, while they may be encountered separately, they only associate with their sister. They are the first lines to have differing classes as well as types, with France an S1 and Holland a BN."

This time the rules were different. France's class of S indicated that she was always lucid (all her beliefs are true), while Holland's class of B meant that she was deranged (all her beliefs are false) during phase B in a three phase cycle, A, B, and C, which progressed to the next phase each week, and lucid for all other weeks. France's type of 1 meant she was deceitful in stage 1, but truthful otherwise; Holland's type of N meant that she was always deceitful.

Two dolls of this description descended before Remilia and her companions. Both were blondes, Remilia idly noted.

83. Lying Dolls

Holland said to Remilia, "I am unreliable today, but I was reliable this day last week."

France added, "I had lied today."

Holland then issued the challenge: "What is our stage and phase?"

Solution: Holland cannot be reliable today, because if she were, then her statement is false which it cannot be if she were reliable. Therefore, she is unreliable, and her statement is false. But she is indeed unreliable today, so therefore it must not be true that she was reliable last week — she must have been unreliable last week too. Therefore, there has been two weeks in a row when Holland has been unreliable, which only occurs when she is lucid, so she will be reliable next week, when she will be deranged. Holland is class B, so next week they will be in phase B, so this week they are in phase A.

France's statement is more subtle. If France is deceitful today, then her statement seems to imply that she has not lied today, despite the fact that she is deceitful today. Yet if she has not lied today, then she is not deceitful — a straight paradox with no way out. However, this only applies if her statement refers to a whole day of not lying, and it doesn't. The statement is in the past tense, so the lies she refers to were before the time that she made her statement. Her statement can be refined to: "I had told a lie today prior to this statement."

Of course, if this is the truth, then she had told a lie today and therefore must be in a deceptive mood, given that she is of designation S1. But then she cannot tell the truth and say that she had lied today, so this statement must be a lie; she has not lied today before that statement. But that doesn't mean she has told the truth today, either. This is her first statement today. So she is deceitful today and the dolls are therefore in stage 1.

—/—

Dolls of Old China

"The sister lines, Beijing and Huizhou, are much like France and Holland, respectively, except the two lines are indistinguishable from each other. They wear chinese cheongsams, yellow and white, and often exchange them like Nancy and Nice. Like the other dolls, they only associate with their sister. Beijing is of designation S1, and Huizhou, BN."

These words echoed in Remilia's head as two dolls in chinese cheongsams dropped down to confront them, clad in yellow and white. Remilia labeled the doll in the yellow cheongsam 'Doll Y,' and the doll in the white cheongsam 'Doll W.'

84. China Dolls

"I am class B," announced Doll Y.

"I am type 1," claimed Doll W.

Doll Y continued, "I believe I am type N."

Doll W then issued the challenge: "Which of us is Beijing, and which of us is Huizhou, and what is our current stage?"

Remilia snorted. To be sure, it was more intricate this time, this challenge, but it still fell to her reasoning.

Solution: From her first statement, if Doll Y is Beijing, then she is currently deceitful, because Beijing is actually class S. On the other hand, if Doll Y is Huizhou, then she is currently deranged, because Huizhou really is of class B, but is also type N, so she can only be reliable if she is in phase B. Either way, Doll Y is in a deceitful mood.

By Doll W's statement, if Doll W is Beijing, then she is currently truthful, because Beijing really is of type 1 and, being class S, is always lucid. If Doll W is Huizhou, then she is currently lucid, because she actually is of type N and always deceitful, and only unreliable when she is lucid. Therefore, Doll W is lucid.

Because Doll Y is currently in a deceitful mood, her second statement implies that she is not of type N. For because she is deceitful, she doesn't really believe that she believes she is of type N. Because she is completely opinionated about her type, she believes that she doesn't believe she is of type N.

Suppose she is lucid. Then that belief is correct, and she doesn't believe she is of type N. Again, this implies she believes she is not of type N, and again, because she is lucid, she really isn't of type N. Suppose instead she is deranged. Then that belief is incorrect, and she actually does believe that she is of type N, and because she is deranged, she isn't of type N. Therefore, Doll Y isn't of type N. She must be of type 1, and furthermore, is Beijing, and Doll W is Huizhou, and because Beijing is currently deceitful, the dolls are in stage 1.

—/—

Twin Cities on Opposite Sides of the World ~ Kyoto and London

"Let's see... France/London... Beijing/Huizhou... ah, Kyouto and London! These dolls are distinguishable, with Kyotos wearing pink kimono, while Londons wear red dresses with white trim and black bearskin hats. One is class S, and the other is class A, and one is type 1 and the other is type 2. I don't know which doll is which, though."

Remilia was amused that she had to consult a cheat sheet for that one. She didn't even need one to recall that a class of A meant that she was deranged in phase A. Of course, every doll had the same cycles of phases and stages

85. The Lion and the Chrysanthemum

A kimonoed doll dropped out of the canopy, obviously a Kyoto. She announced, "I am class A."

The doll in red, London, announced as she settled behind Kyoto, "I believe I am unreliable."

Kyoto added, "London was deceitful yesterday."

London issued the challenge then: "What are our designations, and what are our phase and stage?"

Remilia passed this challenge.

Solution: If London is currently truthful, then she really is unreliable, because a doll that is both truthful and lucid cannot say "I believe I am unreliable." For if she were truthful and lucid, then then she really does believe she is unreliable, but being truthful and lucid, she is actually reliable, and so believes something that is not true — assuming lucidity leads to a contradiction. Therefore, if she is truthful, she must be deranged. On the other hand, if she is currently deceitful, then she doesn't really believe that she is unreliable, and because she is opinionated on her reliability, she believes she is reliable. But the only way she can be reliable and deceitful is if she is deranged, so we have a deranged doll that correctly believes she is reliable, which is impossible. Therefore, she cannot be deceitful. She is truthful, and therefore deranged.

Because London is deranged, she is not class S, but class A. By the same token, the dolls are both in phase A. Also because London is class A, Kyoto is class S and therefore lucid. Kyoto is also unreliable today, as she claimed to be of class A, so she must be in her deceitful stage today.

Kyoto is the deceitful one today, therefore London was not deceitful yesterday and is not today. Therefore, London will be deceitful tomorrow. Since Kyoto is deceitful today, she will not be deceitful tomorrow. Because at least one of them is deceitful today and tomorrow, the day after they will be in stage 3, so they are currently in stage 1. Since Kyoto is currently deceitful, she is type 1 and London is type 2.

—/—

The Favored Dolls of Alice Margatroid

"Finally, we have Shanghai and Hourai, Alice's two favorite dolls. Only one of each of these dolls exist, and of course are sisters to each other. Their types are 1 and 2, and their classes are A and B, but I do not know which doll has which type and which class. Even if I did know, I still wouldn't know which doll is which because they have basically identical body-forms. Unlike the others, their challenge is implicit: to deduce everything about them."

Remilia leaped back as two shapes shot down to raise a cloud of dust with a thunderous slam. "So, we meet again, Shanghai and Hourai," the vampire remarked, seeing the two familiar figures just barely visible through the screen of dust.

Then the two dolls arose, dressed in simple blue Anne of Green Gables type dresses, with white aprons and red ribbons tying back their long, waist length hair. As Remilia had guessed, they were Shanghai and Hourai, Alice's two favorite dolls, and as promised, completely indistinguishable visually.

86. Shanghai and Hourai

The only difference between the two Remilia could see was that one was wearing a pleasant, friendly smile, while the other was wearing an annoyed frown. Remilia mentally and uncreatively labeled the smiling doll 'Doll S' and the frowning doll 'Doll F.'

"What's with the smiling and frowning?" asked the vampire.

The smiling doll —Doll S— said, "Shanghai always smiles and Hourai always frowns."

The frowning doll —Doll F— threw her sister a sharp look. "That's not true!" she protested.

Doll S ignored Doll F's protest. "This day last week, I was deranged."

"You just currently and falsely believe that. I was the one who was deranged last week," countered the frowning doll.

"No, it is you who currently believes false things!" Doll S replied hotly, keeping her smile.

"Enough," Remilia broke in, halting their argument. She then asked of the frowning doll, "Were you reliable this day last week?"

"Yes, I was," she answered.

Remilia tapped her chin. She was able to decipher most of what she needed to know, but there was still a few things she could not figure out from their statements so far.

"You, the one with the frown. Will Hourai smile if I ask her to?"

"If you say 'please,'" Doll F said gruffly.

"Hourai, will you please do so?"

What Remilia saw next enabled her to figure everything out.

Solution: In their first pair of statements, the dolls contradict each other, so they cannot both be reliable or both unreliable. Therefore, they are not both lucid and truthful — one must be either deranged or deceitful. By the same token, it cannot be that one is lucid and truthful and the other is deranged and deceitful, because then they'd both be reliable, nor can they be one lucid but deceitful and the other deranged but truthful, because then they'd both be unreliable.

If Doll S is currently reliable, then she is Shanghai and Doll F is Hourai. For if Doll S is reliable, then Shanghai indeed always smiles and Hourai always frowns. Since Doll S is indeed smiling, she must be Shanghai, and not Hourai who always frowns, and Doll F (who is frowning) Hourai. This fact implies that if Doll F is Hourai and she always frowns, then she is currently unreliable.

Both dolls claim to have been deranged last week, and one being unreliable and the other not, exactly one of them really was deranged last week. Since the dolls are of class A and B, one of the dolls was only in a deranged phase last week, so last week the dolls could have only been in phases A or B, not C. Furthermore, each doll accuse the other of being currently deranged, and again exactly one doll is reliable in her claims, so one really is deranged at the moment. This can only happen in phases A or B, and they cannot be in phase A because that would imply last week they were in phase C, which cannot be the case. Therefore, this week the dolls are both in phase B.

Suppose Doll S is unreliable. Then she was not deranged last week. This makes Doll F reliable, so it is true that the smiling doll only believes she was deranged last week — Doll F blames her sister's current unreliable state on her derangement. The smiling doll then seriously believes that she was deranged last week and is therefore truthful, but deranged. Suppose, on the other hand, the smiling doll is reliable. Then she was indeed deranged last week. This means that this week, the smiling doll is not currently deranged. Since she is also reliable, she is also currently truthful. In either case, the smiling doll is in a truthful mood. By the same token, the frowning doll is also in a truthful mood. Because they are both in a stage that is truthful, the dolls are currently in stage 3. Combined with the previous derivation, the dolls are currently in state B3.

Doll F claims to have been reliable that day one week ago. Today, the dolls are in state B3. Last week this same day, they must have been in state A2, because that day one week ago was seven days ago, which means that their stages have advanced seven times to leave them in stage 3, which means they must have been in stage 2 that day, and their phase has advanced one time to arrive at phase B, so last week they were in phase A. If Doll F is currently reliable, she was deranged and reliable that day last week, which implies that she was deranged and deceitful — she is a doll of designation A2. If Doll F is currently unreliable, then last week she was unreliable too, which means that she was neither deranged nor reliable, which means she was lucid and deceitful — she is a doll of designation B2. Either way, she is a doll of type 2, so Doll S is of type 1.

Remilia was able to figure out all of this before asking Hourai to smile. She could not tell which one was Hourai, nor whether which of the dolls was class A and which was class B.

If Doll S is Hourai, then Remilia's asking Hourai to smile would have no effect. For we already know that if Doll S is Hourai, she is unreliable, which makes Doll F (Shanghai) reliable. This means her statement about Hourai smiling if Remilia asks her politely is true, but of course the consequent is already true, so Hourai (the smiling doll) will continue to smile.

If Doll F is Hourai and she is unreliable, then it is not true that she will smile if Remilia asks her politely — if Remilia asks politely, then Hourai must continue to frown (if she smiles, she is reliable). Since Remilia has asked politely, Hourai would remain frowning.

Either way, if Hourai is the smiling doll, or if she is the frowning doll and unreliable, she will not change her state of smiling or frowning, and Remilia will be unable to tell which doll is Hourai, and bring her no closer to figuring out who is who or what their designations are.

But Remilia was able to figure out everything. Therefore, someone changed their state of smiling or frowning. Hourai cannot be either the smiling doll (because she would continue to smile), nor unreliable (because she would continue to frown), so Hourai must be Doll F, and is now smiling.

This proves that Hourai is Doll F, and that she is reliable. It also proves that Shanghai is unreliable. Since they are both in their truthful state, Hourai (Doll F) is lucid and Shanghai (Doll S) is deranged. Thus, the dolls designations are B1 for Shanghai, and A2 for Hourai.

—/—

\title The Seven Colored Magician

"You did well, Lady Remilia Scarlet," Hourai said, curtseying in midair while still smiling the smile that formed on her lips just seconds before. "Do you have keys with bows shaped like card suits?" In answer, Remilia produced her keys. Hourai nodded at this practical demonstration. Even though Remilia's word was just as good, it seemed to give her a lot more satisfaction to present to the doll material proof. "Two of those keys unlock the front and back doors of Alice's house, one for each door. They must be turned either clockwise or counterclockwise to open the door. One way will unlock the door, but the other way will jam the lock. The direction they must be turned are not necessarily the same."

"Orleans and Lillie are the next-to-last sister lines, the last being Shanghai and Hourai, but more on those two later. Orleans and Lillie are probably the toughest of the dolls to make sense of, because they are completely indistinguishable. They can also be of any designation - and that's not just because I don't know either: every one of their lines run the entire gamut of designations. This is usually of no consequence because the only thing they do is guard Alice's house, and make sure that you not only have the bronze keys with bows shaped like card suits, but also that you use the right key on the right lock. However, I do know that no pair of sisters have matching type or class."

"You know, we haven't met an Orleans or a Lillie doll yet," said Remilia aloud.

"That's because the only pair about is currently guarding Alice's house," Hourai replied. The doll saw their curiosity piqued, and continued, "They know everything there is to know about Alice's locks —which keys to use on each lock, and which direction they must be turned— at least when they're lucid."

Alice's doll took a moment to scrutinize Remilia and company, scratching her chin out of seeming habit. "Yes, I think you'll do nicely," she said, mostly to herself. Then she started long the path again, bade the four to follow. "Come. I shall lead you to Alice's home. With me as your escort, you will encounter no more challenges from other doll patrols."

87. Alice's Doors

As promised, with Hourai as escort, the troupe was unharassed as they made their way to the Margatroid manor. Remilia could catch glimpses of fairy dolls peeking out briefly before scurrying away. Soon, they came onto the clearing where Alice's house lay. As promised, a pair of Orleans and Lille dolls floated before the front entrance, armed and armored, and with a serious look on their faces.

"I feel that there is no trap."

Remilia nodded at Sakuya's words. "Assuming you haven't gone insane already, we should be cautious," Remilia agreed to the implied truth behind Sakuya's lie. The vampire turned to Hourai. "Should we approach from the front or the back door?" she asked.

"Alice is waiting behind one of these doors. If you open the wrong door, then Alice will put a stake through your heart."

Remilia swallowed. A stake through her heart wouldn't kill her, but it would incapacitate her, allowing the mad magician to do other unspeakable things to her.

Honey fluttered in front of Remilia, blocking her progress temporarily. She added an important point to her advice, "I must caution you also that this will be quite a puzzle to solve. I suggest taking notes."

Remilia felt a touch of annoyance at the fairy's sudden imposition, but relented. "Right. Sakuya, your notebook, and use your truthful hand," she commanded her maid.

The maid flipped out a pad of paper and a pen, holding it in her left hand. Her truthful hand. "I shall not do so, m'lady," lied the maid.

Remilia gave her servant a nod, then stepped forward.

The two dolls immediately reacted, crossing their lances in front of her, as if daring her to cross swords with them. Remilia's only reaction was to hold up her four keys. The dolls were suspicious, but were at ease. "State your questions, stranger," ordered the doll wearing blue.

Remilia ignored the harsh tone of the doll. "First, please tell me a bit about yourselves," she started.

"We are not both currently unreliable."

Sakuya quickly jotted down the statement of the doll bedecked in red finery. Remilia took a moment to peek over Sakuya's shoulder, to see that she had helpfully labeled Red's first statement as (Red-1). She nodded at the red doll to continue.

"In addition, my sister is neither class S nor class A."

Remilia saw this statement was labeled (Red-2).

"Finally, Orleans is wearing blue," the red doll finished.

Remilia guessed this would be labeled (Red-3). A quick check confirmed this.

The blue-clothed doll then began to speak. "I am not both lucid and deceitful. I am also not of type 2," she began.

Remilia peeked over Sakuya's shoulder again, and saw those statements labeled (Blue-1) and (Blue-2) respectively. Remilia noted that not a single word was left off by her maid's scribbling. No shorthand. The words were written as they were spoken. Remilia felt momentarily proud.

"As to our classes, Orleans is of class S and I am class B. One of us is not of type K. As to our current phase, we are now in phase B," added the blue-dressed doll.

Remilia checked to see that these three statements were labeled (Blue-3), (Blue-4) and (Blue-5) respectively.

Remilia waited a few beats for any other forthcoming statement, but the dolls remained silent. It seemed they had stated all they wanted to say about each other. Time for her to move on. "Okay, what can you tell me of the locks?"

Again, the red doll spoke first. "Either you have used one of the correct keys before on a previous lock, or the correct keys have the same jewel set in their bows."

(Red-4), thought Remilia. By this time she trusted Sakuya could keep up with the transcription.

"The keys must be turned the same way in their respective locks."

(Red-5).

The blue doll broke in this time. "I believe that turning the spade-bowed key counterclockwise will not jam at least one of the locks."

(Blue-6), thought Remilia, still keeping count of the statements in her head.

The red doll followed up with a quite similar statement. "I believe that turning the spade-bowed key clockwise will jam both locks."

(Red-6), thought the vampire girl.

The blue one spoke again. "If you had asked me this day last week, I could have claimed that the correct key for the back turned counterclockwise. If you ask me again this day next week, I could claim that the correct key for the back must be turned clockwise."

(Blue-7) and (Blue-8).

"If the front door lock takes a red-jeweled key, then the back door lock also takes a red-jeweled key," came Red's final statement on the keys and locks of Alice's abode, which Remilia knew would be labeled (Red-7).

Again, the silence was unbroken for several beats, indicating the dolls would say nothing more on the subject. The vampire glanced towards her maid, who was furiously scribbling away on the notepad, trying to commit the dolls' words to paper. Her brow was glistening with sweat, and her mouth was pursed tight. After she finished with the transcription, she sighed and wrang out her left arm. While that arm may be as skillful and legible as her dominant arm, it wasn't actually used to lots of writing.

Remilia decided to wrap this up. "I know that Alice will stab me through the heart if I open the 'wrong door.' Please tell me which door is the correct door."

"We are both reliable today if and only if the front door is the door that is not 'wrong,'" said the red one.

Remilia glanced towards Sakuya, and saw her complete transcription of the statement that the vampire knew to be (Red-8), if she got her statement count right. Her left arm was now looking very pained, and Remilia held up her hand. "I think we'll be fine without any more transcription, Sakuya." Turning towards the dolls, she asked one last question, "Red one, what class do you believe yourself to be?"

"I believe I am class I," came the doll's answer.

Remilia took the notebook off Sakuya's hands. It was written in her maid's neat handwriting, the writing of this knightish hand indistinguishable from the handwriting of her actual dominant hand. She scanned the notes, trying to fit the logic together. It took a few minutes of working through the logic, but she was finally able to truthfully announce, "Orleans, Lille, I thank you. I have figured out, amongst other things, which key goes to which lock, which direction they should be turned in, and which entrance to use."

Solution: The first step is to establish the reliability and identity of both Red and Blue.

If Red is reliable, it's certainly true that they are not both unreliable (Red is reliable), and thus, her statement (Red-1) is the truth. Also, Red really isn't Orleans, because Red is not wearing blue; her sister is (Red-3). On the other hand, if Red is unreliable, then those two statements imply that it is not true that not both dolls are unreliable, so they are both unreliable, and also that Red actually is Orleans because the untruth of her statement implies that Orleans is not wearing blue, so she must be wearing red. Therefore, if Red is unreliable, Blue is as well; and Red is Orleans if and only if she is unreliable.

If Blue is reliable, then she is not Orleans and is therefore Lille, because she claims Orleans and her are of differing class (Blue-3). But this means that Red is unreliable, because she claimed that Orleans is wearing blue, and Red's unreliability implies Blue's as well. So we have reached a contradiction: if Blue is reliable, she is not reliable. Therefore, Blue is unreliable. Since Blue is unreliable, she is both lucid and deceitful, and of type 2 (Blue-1 & Blue-2). Therefore, she is not class I. This means she can be class S, A, or B, and of type 2. Blue disclaimed that one of them is type K (Blue-4), so we know that one of the dolls is indeed of type K. Since we know Blue is type 2, we know Red is of type K, and is always truthful (though not always reliable).

Red is truthful, so her answer to Remilia's last question reveals that she believes that she believes she is of class I. Suppose Red really does believe she is of class I. Then she is reliable, because she has stated something that is true. Furthermore, she believes something that is true by the same token, so she is lucid today: she really does believe she is class I. Consequently, because she is lucid and believes she is class I, she really is of class I. But this cannot be, because class I dolls are always deranged. So if she really does believe that she is of class I, then she is both lucid and deranged at the same time, which is impossible.

She cannot believe that she is class I, contrary to her statement, so she is unreliable. But we have already shown she is truthful, so she must be deranged. But she is completely opinionated about her class, so if she doesn't believe she is class I, she must believe she is not of class I. But this is false because she is deranged. Therefore, she must be of class I.

In addition, Red's unreliability leads us to the conclusions that Red is Orleans, Blue is Lille, and they are both unreliable. We've established Red and Blue's identities and reliabilities, and can now move on to the keys.

Since Orleans is not reliable, then it is not true that one of the correct keys has been used in a previous lock, nor is it true that the correct keys have the same jewel set in their bows, contrary to her statement (Red-4). Therefore, there is one correct key that has a red jewel, and one correct key that has a black jewel. Furthermore, the correct key that has the red jewel cannot be the diamond-bowed key (which was used in a previous lock), so it must be the heart-bowed key.

Orleans and Lille know everything about Alice's locks while they are lucid. In particular, they know (when lucid) whether or not the spade-bowed key will jam which locks if turned clockwise or counterclockwise. This makes them opinionated on these positions. When deranged, they are still opinionated on these direct statements of fact.

Orleans is truthful, so her statement (Red-6) means she believes that she believes that turning the spade-bowed key clockwise will jam both locks. But she is deranged, so she in fact does not believe this direct fact. However, she is opinionated on it, so she believes that this direct fact is not true, and again because she is deranged, this direct fact is indeed true: turning the spade-bowed key clockwise in either lock will jam that lock. (Note that if Orleans were lucid, the same result would hold: turning the spade-bowed key clockwise in either lock will jam that lock.)

Lille is deceitful, so her statement (Blue-6) means that she believes that she does not believe that turning the spade-bowed key in at least one of locks will not jam it if turned counterclockwise. Since she is lucid, she really does not believe this direct fact, and since she is opinionated on that direct fact, it implies that she believes this direct fact is not true. Again, because she is lucid, the direct fact actually is not true: the spade-bowed key will jam both locks if turned counterclockwise. (Note that if Lille were deranged, the same result would hold: turning the spade-bowed key counterclockwise in either lock will jam that lock.) Thus, the spade-bowed key will jam the lock if turned clockwise or counterclockwise: the spade-bowed key does not fit in either lock and may be ignored hereon. Therefore, the club-bowed key is a correct key.

Lille is now in stage 2, because she is a type 2 doll and currently deceitful. Last week this day, Lille was in stage 1 (seven days in the past), so she had been truthful that day. She will be truthful again a week from now (seven days hence) because she will then be in stage 3. She is unreliable today, so actually, she could not have claimed that the correct key for the back would be turned clockwise and next week she could not have claimed that same key would need to be turned counterclockwise (Blue-7 and Blue-8). Since Lille is truthful at both these points, the difference must be because she is lucid at one of these points and deranged at the other point. Either last week she was deranged, or next week she will be deranged (she is currently lucid). Orleans' unreliable statement (Red-2) claiming that Lille is neither class S or A implies that, actually, Lille is either class S or A, but if she is of class S, then Lille cannot change from lucid to deranged. This means she must be of class A.

Lille's statement (Blue-5) implies the dolls aren't really in phase B this week. Lille is lucid this week and she is of class A, so they are not in phase A this week. They must therefore be in phase C, and next week they will be in phase A. Lille will be deranged that week, so next week this day, she will be truthful and deranged, so she will be unreliable. Therefore, the fact that she could not claim that the correct key for the back should be turned counterclockwise is because that would be the truth. So the key to the back door must be turned counterclockwise. By Orleans' unreliability and her statement (Red-5), the keys must be turned opposite ways. Since the back door key must be turned counterclockwise to open it, the front door key must be turned clockwise.

Orleans statement (Red-7) is false, so the implication is false, so its antecedent is true while its consequent is false. This means that the front door takes a red-jeweled key and the back door takes a black-jeweled key. The only red-jeweled key that can be a correct key is the heart-bowed key, while the only black-jeweled key in the running is the club-bowed key. Similarly the whole logical equivalence of statement (Red-8) is false, and so the two are both reliable today if and only if the front door is not the door that is not wrong. However, because Orleans (and indeed Lille) is unreliable today, the antecedent is also false, so it must be the case that the front door is not the wrong door.

—/—

Remilia unlocked and threw open the front door, confidently.

*THUNK*

"M'lady!" shouted Sakuya in surprise, shock and dispair.

Remilia was just then feeling the shock of something impacting her breastbone, even as her brain processed the fact that a dark shape had lunged at her as soon as she opened the door. Remilia could only look down and dumbly note that a wooden stake was protruding from her breast, probably piercing her heart, driven firmly and all at once into her sternum by the deceptively willowy arms of Alice Margatroid. Alice's face bore a look of crafty and slightly sneering satisfaction.

The Seven Colored Magician had staked her, she realized, and had been waiting for her at the front door.

"What?" she said, her voice wavering as the strength drained from her, "But... I thought... Hourai..."

"She did," replied Alice, "But she didn't say that if you chose the correct door, that I would not stake you through the heart."

"Bugger," the staked vampire managed to croak out before darkness claimed her.

—/—

Remilia awoke to find herself seated in a chair, bound by the preternaturally strong filaments of Alice's puppet strings, puppet strings held taut by quintuplet of Orleans and Lille dolls. She was not sure if they were the same dolls as were guarding Alice's house when she and Sakuya arrived and she was unsure of how much time had passed since she had been staked.

Speaking of Sakuya, she was across the room from her, in heavy wooden stocks and chained securely by tight manacles on her wrists and legs. Even if she were to activate her time-manipulation powers, it would avail her naught — the manacles would hold her just as securely in her timeless space as they do in this temporal space.

"Sakuya!" shouted Remilia. The Perfect, Elegant Maid lifted her head to face her mistress, and Remilia saw that she was less than perfect, less than elegant. It almost seemed as if several days had passed.

"Mistress... You have revived before today..." groaned the maid.

Remilia immediately deciphered her lie. It strongly implied that she had spent at least a day, perhaps several, unconscious. "Sakuya! Are you well? What did Alice do to you?"

"I am well-fed, or well-rested," Sakuya said weakly, her face slightly gaunt and wearied. "I have been given an inadequate amount of potable water, though. As to the rest, as of now, she has done something other than give me water."

"Is this still the same day, Sakuya?" Remilia asked, glancing at the light of the setting sun.

"I do indeed know whether this day is the day we arrived," replied Sakuya, which Remilia took to mean she did not in fact know.

A loud clack of the twisting of a doorknob announced the arrival of their ultimate captor, Alice Margatroid. An unfriendly smile graced her lips, and upon a metal tray she carried three wicked looking syringes filled with sickly-green serums.

Remilia wasn't about to be intimidated by a bunch of syringes, like a little child. "What are those, Seven Colored Magician?" she asked defiantly.

"Something whose significance I will soon demonstrate." With that, Alice turned around and walked out of the room, closing the door behind her. A very brief time past, probably only a minute or so, before she returned to the room, this time carrying the white-bedecked form of Lily White, bound and gagged and struggling like an earthworm plucked out of the ground by a crow. "To each of my victims I pose a challenge. If she answers correctly within a time of fifteen minutes, I set her free, unharmed and without alteration. If she fails to guess, or guesses falsely, I administer this serum." Alice raised Lily White's writhing and distraught form and added, "This child has failed the challenge, and she will suffer the consequences."

Alice laid the struggling fairy upon a small table and strapped her down. She then picked up one of the syringes, and roughly jabbed it into an arm laid bare. Lily mewled in agony through her gag as the gunk was injected into her system. The fairy's eyes unfocused, then glazed over as the serum took its sinister effect. Within minutes, the fairy herald's eyes cleared once more, but Remilia didn't think that this meant that she was all right again.

Alice released the fairy, taking the gag off. "Now, Lily-dear, are you a knight or a knave?" she asked.

Lily smiled and declared, "I'm a knave!"

A cold chill ran up Remilia's spine. She knew what significance that statement had for Lily's sanity. "She's insane now!"

"You are correct," Alice replied. She then launched on a typical B movie villain explanation, though Remilia didn't really know what a B movie villain was, she certainly recognized the cliché. "I have been experimenting with lucidity and derangement these past months before the incident started. The Great Troll commissioned me to do research into knighthood and knavehood with my dolls, as well as these peculiar states. You have encountered the fruits of those experiments. Yukari decided to forego the dimension of lucidity and derangement at large, but it intrigued me to no end. Through my experiments, I developed this serum. It destroys its victim's sanity, flopping all of their beliefs and rendering their victims permanently deranged... insane."

"Like your dolls do as a matter of course, when they switch from lucid to deranged from week to week. The only difference is that the dolls can flip back," Remilia observed.

Alice looked pleased at Remilia's observations. "Correct again, Remilia Scarlet." The Seven Colored Magician looked contemplative for a moment, "I tell you, I had my work cut out for me converting half the forest to insanity in six days."

Remilia felt an even colder chill run down her spine. "Then... all of the insane beings in this forest-"

"Are my doing, yes!" finished Alice Margatroid in triumph.

88. The Sanity of Alice Margatroid

"You're mad!" growled Remilia.

"But am I insane?" Alice asked rhetorically, following it with a chuckle. That was the question, wasn't it? Madness and insanity didn't mean the same thing in this forest as they did outside.

Remilia narrowed her eyes. "I know that you're reliable. Otherwise, you would have denied the obvious truth that Hourai had said that you would put a stake through my heart if I chose the wrong door and that she did not say you would put a stake through my heart if I chose the right door. So you do indeed speak truth, regardless of whether you mean to or not," she observed coldly.

"You are quite correct, Remilia Scarlet. I guess all you need to find out is whether I am a knight or a knave, but you won't get that from me. It's too easy," Alice chuckled.

Remilia frowned, puzzling over how to figure out if Alice was insane or not. She was about to probe for more information when she spied Honey Humblebee out the window, waving to her. As her eyes focused on her, she immediately slapped a note on the window, written in big letters (albeit a bit sloppily) reading, Alice's sanity has been altered exactly once, or I have written this note with my knaved hand! —Honey Humblebee

Alice heard the noise, and turned around to see what had collided with her window, but the sign was already gone by the time she looked, which lead to Remilia only barely having enough time to remember what was written upon the note herself. Honey made a face in mockery of the magician, then buzzed off.

"Ugh! I've been trying to capture her for ages, ever since my last experiment on her!" Alice huffed, glaring out the window.

"It doesn't matter. I've already figured out you're insane," said Remilia, glaring at Alice.

Solution: Honey is, of course, sane and omniscient, so she either believes her written statement or its negation, and her belief will be correct. This means Honey's knighted hand is reliable and her knaved hand is unreliable. If she wrote the statement with her knaved hand, then the statement is true, which cannot be because her knaved hand cannot write true statements. It must be written with her knighted hand. But if her note was written with her knighted hand, it certainly wasn't written with her knaved hand, but the statement must be true, so Alice must have had her sanity altered exactly once.

All the insane beings of the Forest of Magic are the work of Alice Margatroid. Therefore, she must have been sane before her sanity had been altered, and thus insane now.

—/—

"Indeed, I am insane! In fact, I am the first being of the forest to attain the blessed state of insanity," Alice announced boisterously.

"But you started out as sane," Remilia pointed out.

"And I would have remained so if by chance I had not accidentally dosed myself with my own serum. Ah, to be free of sanity's shackles, such a blessed feeling. Now, you and your faithful —if fibbing— maid shall be free of sanity too!"

If Remilia didn't already know Alice was insane, she would have believe she was making stuff up. It was if she was acting in a primary school production taught by Ms. Keine, only Alice was serving up great slabs of ham. She could feed the entire human village off the ham that Alice was serving up.

Nevertheless, Remilia felt the need to counter Alice's nonsense. "You call it 'shackles,' I call it 'grounding.' I've never before been so envious of a doll, you knavish nutter. At least some of them are lucid part of the time."

"Do not trivialize your situation, Remilia Scarlet! Your jest is not funny!" Alice exploded in rage. Remilia silently answered that it wasn't intended to be. "Any more of that, and I will not even give you the usual chance to escape your fates!"

"So you were saying," Remilia said, dropping the previous thread of conversation. "It seems rather sporting and generous of you, Seven Colored and Insane Magician. Why do you do this?"

Alice blinked quizzically. "Does it seem so? I'll have you know that I have not a generous or sporting bone in my body. I derive an enormous sadistic... pleasure... watching my many victim squirm and worry as they flit about the air impotently, and scribble desperately trying to figure out the answer. It is more than fair compensation for the infinitesimal chance that they may solve my challenge."

Remilia well noted Alice's use of 'infinitesimal;' it offered no comfort whatsoever — quite the opposite. "You suppose correctly, Remilia Scarlet: I have not lost a victim yet. I run very little risk," concluded Alice.

"I have no choice anyway. What is your challenge?" asked the bound vampire.

89. The Challenge of the Mad Magician

"You see, there is a statement S that has an almost magical property. If there is any other statement X whose truth or falsehood you want to ascertain, all you would have to do is ask someone who is opinionated on X, 'Is S equivalent to X?' If you get 'yes' for an answer, X must be true; if you get 'no', X must be false.

"For instance, 'Is S equivalent to the statement that you are a knight?' and she answers 'yes,' then that person is a knight. If you were to ask her 'Is S equivalent to the proposition that you are sane?' then if she answers 'no,' she is insane. All provided they are opinionated on their knighthood and sanity, of course."

Remilia was very curious about the identity of this sentence. "What is this sentence S?" she asked in genuine curiosity.

Alice's smile tightened, amplifying its nastiness. "Ah, that is for you to deduce! It is your challenge!" she said, bopping Remilia's nose with her finger. Remilia nipped impotently at the digit, but Alice quickly removed it from harm casually. "You have fifteen minutes. You'd better think hard; the stakes are quite high," she stated, stepping back and then through a waiting door.

"Quite high, indeed..." Remilia growled as the door closed behind Alice. So worried for Sakuya's sanity and her own, that Remilia's mind was almost paralyzed with the prospect, and was quite certain that Alice, that sadistic and insane Seven Colored Puppeteer, was watching from her secret hiding place.

Fifteen minutes elapsed, whereupon Alice entered again. There was a wide, sick grin on Alice's face, speaking of her insanity within. She deliberately picked up one of the filled syringes, and moved over to Remilia.

The vampire instinctively tried to get away from Alice, knowing that the syringe contained a substance that would turn her insane, in this special sense, and was repulsed by the idea that she would become a creature that, no matter how normal she seemed on the surface, believed only that which was false. Then, when Alice was almost about to jab the needle into her arm, Remilia's mouth suddenly broke out into a wide grin of realization as she shouted, "Of course! The sentence is..."

It was this sentence that saved her and Sakuya.

Solution: Many propositions would work, but one possible S is, "You are currently reliable."

Suppose the subject is (currently) reliable. If X is true, then the subject's reliability is equivalent to the truth of X, and being reliable, she answers "yes." On the other hand, if X is false, then the subject's reliability is not equivalent to the truth of X, and because she is reliable, she answers "no."

Now suppose the subject is (currently) unreliable. If X is true, then the subject's reliability is not equivalent to the truth of X, but being unreliable, she would answer falsely with "yes." On the other hand, if X is false, then the subject's reliability is equivalent to the truth of X, and being unreliable, she would answer falsely with "no."

In either case, if X is true, then the subject answers "yes," and if false, the subject answers "no."

—/—

For a crucial moment, the threads that held Remilia slackened with her shock. The vampire capitalized on this opportunity and lunged at Alice, knocking the syringe into a corner of the room and pinning the youkai magician to the ground.

"Never before has one managed to solve my puzzle and escape with their sanity intact," growled the magician.

"Then it pleases me to be the first," Remilia replied with a sneer. "It could have easily been my fate. I was rather scared out of my mind. But that's over now, Seven Colored Magician." Remilia curled back her lips, exposing her wicked dental apparatus. "Use your dolls to free my maid. Now," she hissed menacingly.

Alice looked almost as if she would not, but relented with a sigh. The Lille and Orleans dolls moved to release Sakuya from the heavy stock and release the manacles holding her fast. Sakuya extracted herself groggily from the stocks, swaying a bit, both tired and drained. Steel returned to her figure, however, when she flicked out a fan of her knives. "On your word, M'lady, I shall not begin to make her pay for her affront!"

"I do not give the word as of yet, Sakuya," Remilia snapped back, keeping her blood-red eyes on Alice. "Anyway, if you somehow managed to capture Honey Humblebee, or another one of a certain class of fairies, they would have been able to answer your challenge immediately," added Remilia after seeing Sakuya move to extract herself from the stocks and manacles that held her. Seeing Alice's confused look, Remilia teased her with a hint, "They're opinionated and sane — think about it, and you may figure out why." Remilia paused, thinking over what she'd just claimed. "Or maybe not. Your beliefs are screwed up at the moment."

"Now that you have me, Remilia Scarlet, what will you do with me?" asked Alice. "Will you give the word to your servant to extract your revenge out of my hide?"

Remilia thought about that for a long moment, staring into Alice's defiant eyes. Then she smiled, deciding against that. After all, Alice was not afraid of her despite her current disadvantage, and Remilia always respected such iron will. "No," the vampire said at last, "Instead, you will say to me that you will not inflict your insanity onto anyone ever again, Alice Margatroid."

Alice narrowed her eyes in annoyance. "Don't make a sane knave out of yourself, Remilia! I believe you don't know for certain that I will make such a statement!"

"That statement works against you, whether you realize it or not. I have seen and manipulated your threads of fate. You will make that statement. I guarantee it," Remilia hissed through a tight smile. "Say it, and be bound by the power of Gensokyo's logic, for you already are."

Alice narrowed her eyes at Remilia. The vampire didn't know what the insane magician was thinking, but she knew Alice's fate. Indeed, she would capitulate right about... now! "Fine. I will not inflict insanity onto anyone ever again," said the Seven Colored Magician with a sigh. She, of course, did not believe that she stated this promise, but it did not matter. She was reliable, and consequently as good as her word. Remilia breathed out a breath she did not realize she was holding.

Just then, one of Alice's favorite dolls, either Shanghai or Hourai, burst into the room, tackling Alice in a hug as she wailed into her bosom. "Master! I'm so happy that you put a stop to this nonsense!" she mewled.

"Hourai?" asked Alice, confirming the doll's identity.

The doll stared up at Alice with dewy eyes. "It isn't right to inflict insanity on people against their will, Master. I'm so glad you said you would stop turning people insane! I turn insane for one week out of every three. I do not find it pleasant when I return to my senses a week after," she babbled.

"While I believe madness is good for you, I shall abide by my words," replied Alice. Hourai giggled and hugged her tighter.

Remilia noted that not only was it true that Alice believed that madness was good for you, and also noted that, being insane, that belief is false. She wondered if she would ever come to appreciate that fact. (Obviously, not until she was sane again.)

"Hourai, how long have we been captured?" Remilia asked.

"It is now morning of the day after," Hourai said.

Remilia knew that Hourai was still reliable due to her quite true statement that she spent every third week deranged, which was true, so she was reliable today. Which means that it was now the next day, as she just said. "Well, it's not as bad as I feared, as we haven't been gone for days and days, but I do think you owe me, my servant, and Lily White breakfast and an apology."

"So be it. I apologize for my mistreatment," Alice said with a bow. "Hourai, please prepare some breakfast for me and our guests."

"Yes, Master!" Hourai said in turn, and fluttered off to prepare some repast.

90. Curing Spring's Herald

After a hearty breakfast of pancakes, Remilia was browsing through Alice's stuff for a potential cure for at least Lily White and Letty Whiterock. Alice insisted that she had no general cure, saying that she had no interest in making such a formula. Remilia ignored this, brushing her off easily with a comment that one of her formulas might be able to restore Letty and Lily in particular, even if she did not have a general cure.

Remilia spied a formula that caught her interest. "'Experimental Formula #32: Opinionation Serum.'" She took a vial of the yellow formula out of the felt-padded case it was stored in with several others. "What's this about, Alice?" she asked, presenting the vial.

"A failed formula," Alice said, somewhat morosely, "I was trying to bestow blessed insanity with that serum, but instead induced opinionation. The subjects I used it on turned opinionated instead of insane. Honey Humblebee was one of them."

Remilia turned that fact over in her head, even as she turned over the vial of Opinionation Serum in her hand. Suddenly, her eyes widened in realization. "Alice," she brusquely called to the mad magician, "will this serum cancel out Letty's or Lily's status as seasonal spirits?"

"No," replied Alice.

Remilia smiled. She left the box she extracted the samples from two vials short. She handed one sample off to Sakuya. "Perfect. Sakuya, take this and go find Letty Whiterock. I shall administer this dose to Lily. Lily?"

"Miss Scarlet?" Remilia's vampire speed guided the syringe to Lily, injecting her with the sample before she could yelp in pain, "OW!"

Remilia waited as Lily rubbed her sore bottom, watching her blink and straighten out in wonder, eyes clearing.

"Lily, are you a knight?" asked the vampire.

Lily smiled and shouted in the joy of realization, "YES!"

Remilia smiled. This was indeed a cure to Alice's induced insanity, at least in seasonal spirits.

Solution: In a previous problem, Lily's statement (prior to being made insane) that seasonal spirits believe that it is exactly one season at any one time implies that such a spirit is opinionated on every season only if she is sane. For if the spirit is insane, then (without loss of generality) she believes the season is currently winter when it is actually currently spring. Because she only believes it is one season at a time, if she believes it is currently winter, she does not believe it is fall. But if she is opinionated on the seasons, if she doesn't believe it is fall, she believes it is not fall. But by hypothesis, it is indeed not fall, but spring! This belief is correct, which cannot be if the seasonal spirit is insane. Therefore, if she is opinionated about the season, she cannot be insane, and must be sane.

The serum alters seasonal spirits such that they are opinionated without altering the fact that they believe that it is exactly one season at a time. Therefore, if they take the serum, Letty and Lily will continue to believe only one season is the true season, and at the same time be opinionated. If they are opinionated, then they are opinionated about the season, and therefore, by the above reasoning, they will become sane.

—/—

Remilia and Lily found Letty Whiterock milling not far from where they first encountered her, freezing a stream within line of sight with the boulder she had been sitting on. Sakuya had just finished injecting the serum, and pulled the needle from her arm. Soon, Letty's eyes cleared, the serum taking effect.

Lily smiled and threw her arms out wide. "It's good to have you back, Letty!" she said cheerfully.

"It's good to be back, Lily," said Letty calmly, "And I'm glad you're back, too."

"Thanks!"

Remilia could feel the sweetness rotting her teeth. "Okay, okay... enough..." murmured the vampire. "We must go now. I only hope that somebody solves this Incident before the turn of the seasons." With that, Remilia and Sakuya took to the air, heading back to the Scarlet Devil Mansion.

Letty smiled, staring after them as they flew. "You will, Remilia Scarlet. I know it!" she whispered.

—/—

Interlude: Smullyan's Treatment

A loud roar of outrage echoed through the Scarlet Devil Mansion. The roar shook the mansion from foundation to roof, and was followed by the quieter but no less audible stream of curses flowing liberally from the mistress's mouth, in a mix of European languages.

The roar startled Flandre out of her concentration, as the clay training sphere she held in her hands popped, her power going out of control and shattering it. "Woah! What's my big sis so steamed about?" asked the blonde loli vampire, suddenly curious.

Patchouli's mouth stretched into a wide, thin smile. "If I had to guess," said the unmoving library magician with a lilting, playful voice, "then she's just discovered Smullyan's treatment of vampires."

"WE DO NOT ALL LIE, SMULLYA~~~~~N!"

(To be continued... unless I am a doll!)

There will be at most six more chapters, owing to the increasing difficulty of the puzzles I'm constructing, and the fact that I only have so much material. I might have to cut down on the number of puzzles per part to keep them manageable. I hope to never again have a puzzle where one has to keep track of multiple statements.