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Seems to me the force needed to penetrate tracks the diameter, but the strength tracks the area of the cross-section.
That is, decrease the thickness by N and it decreases the force needed by N but the strength by N squared.
Below a critical thickness, the wire would just break.
Spiderwebs don't slice you up if you run into them.
I think it's pretty obvious.
Voldemort has always been attracted to power, and it's well known that Hermione is the most powerful witch of her generation.
He made several overtures to her, but was unable to turn her from her path, and so he killed her.
Upon her death he felt great remorse (such was his passion) and decided to bring her back from the dead (such was his power).
Dumbledore tried to stop him, and so was eliminated.
In fact, Voldemort was so enamored of Hermione, that after she was brought back, he use dark magics to give her even greater power.
Quirrell (who has been hiding his identity of David Monroe) was secretly on hand for the ceremony, but by the time he realized what was happening, it was too late to stop it.
Cutting charms were used on Voldemort's hands, and other terrible damage, but despite all this, Quirrell was defeated.
Ironically, having given her the power of friendship, it was the power of friendship which ultimately was his downfall.
I notice you are confused. I think you've made two questionable assumptions;
Assumption 1. Wizard Children are not generally treated as competent at age 11.
Assumption 2. The children making the announcement at Hogwarts are responsible for brokering the deal. I.e. they aren't just mouthpieces for their respective families.
Assumption 2b. The Hogwarts staff is aware of 2.
Assumption 1 might be true - but I note that the age of majority has been increasing over time, and wizarding society is in many ways old timey. It seems reasonable to me that allowing a child of 12 to command a wizarding army is no stranger in wizard society than allowing David Farragut to command a ship at age 12 was during the war of 1812. Also, we haven't seen the reactions of the wizarding world in general - maybe everyone who isn't on the Hogwarts staff is scandalized. For that matter, maybe the staff is too, they're just not openly scandalized.
But assumption 2 seems completely wrong to me, and likely the main source of the confusion.
True Patronus couldn't look like a snake.
I see no justification for that statement. Perhaps True Patronuses can't take the form of an animal, but that says nothing about what they can look like.
Would a sentient snake wizard say a True Patronus can't look like an ape?
Well, it's probably supposed to be spelled "Momroe" as in "David Troll Momroe". :)
It's spelled "Monroe" in Chapter 86, and there's a "Most Ancient House of Monroe". Personally, I never get these names right either, but I keep a text file handy with all the names, and hard to spell spells like Legilimency Occlumency Occlumens, and Legilimens. Then it's just a simple matter of cut and paste.
Quirrell's dash to the scene ... indicate that they are afraid of what this experience will do to Harry.
It seems more likely to me that Quirrell's dash was primarily for the purpose of burning holes in Hogwarts. Despite leaving before Harry, and Harry stopping to pick up the twins and stopping at the library, and supposedly making a more direct route, Quirrell still failed to arrive before Harry, or for that matter, at all.
I'm not saying Quirrell is unafraid of what this experience will do to Harry, just that I don't believe Quirrell's dash is evidence of that.
There's a scam I've heard of;
Mallet, a notorious swindler, picks 10 stocks and generates all 1024 permutations of "stock will go up" vs. "stock will go down" predictions. He then gives his predictions to 1024 different investors. One of the investors receive a perfect, 10 out 10 prediction sheet and is (Mallet hopes) convinced Mallet is a stock picking genius.
Since it's related to the Texas sharpshooter fallacy, I'm tempted to call this the Texas stock-picking scam, but I was wondering if anyone knew a "proper" name for it, and/or any analysis of the scam.
There is a contradiction here between "lucky" and "coin flip". Why does he get lucky on Earth?
I don't see the contradiction. C-Omega tries the same con on billions and billions of planets, and it happens that out of those billions of trials, on Earth his predictions all came true.
Asking why Earth is rather like asking why Regina Jackson won the lottery - it was bound to happen somewhere, where ever that was you could ask the same question.
In the original problem Omega runs a simulation of you, which is equivalent to T-Omega.
I could not find the word "simulation" mentioned in any of the summaries nor the full restatements that are found on LessWrong, in particular Newcomb's problem. Nor was I able to find that word in the formulation as it appeared in Martin Gardner's column published in Scientific American, nor in the rec.puzzles archive. Perhaps it went by some other term?
Can you cite something that mentions simulation as the method used (or for that matter, explicitly states any method Omega uses)?
Consider the following two mechanisms for a Newcomb-like problem.
A. T-Omega offers you the one or two box choice. You know that T-Omega used a time machine to see if you picked one or two boxes, and used that information to place/not place the million dollars.
B. C-Omega offers you the one or two box choice. You know that C-Omega is con man, that pretends great predictive powers on each planet he visits. Usually he fails, but on Earth he gets lucky. C-Omega uses a coin flip to place/not place the million dollars.
I claim the correct choice is to one box T-Omega, and two box C-Omega.
Can someone explain how it is in the “original” problem?
That is, what mechanism does the “real” Omega use for making his decision?
on 3; From chapter 6
As his hand touched the back door's handle, he heard a last whisper from behind him.
"Hermione Granger."
"What?" Harry said, his hand still on the door.
"Look for a first-year girl named Hermione Granger on the train to Hogwarts."
"Who is she?"
There was no answer, and when Harry turned around, Professor McGonagall was gone.
Seems clear to me that the whisper came from McGonagall - Harry was talking to her, Harry turned his back and heard a whisper from "her" that sounded like her. Harry thinks so to - in chapter 8 we have;
The boy's mouth was hanging open. "Were you told to wait for Harry Potter on the train to Hogwarts, or something like that?"
"No," Hermione said. "Who told you about me? "
"Professor McGonagall and I believe I see why.
Doesn't follow. Consider;
I claim a rock is a non-person.
I expect you accept that statement, I expect that you therefore have a non-person predicate function, yet I also expect you haven't found the answer.
I accept that in order to classify something, we need to be able to classify it.
I'm suggesting there might be a function that classifies some things incorrectly, and is still useful.
Yes, that was sort of the point - you can't make a function for "is a Turing machine" that works in all cases, and you can't make a "is a non-person" function that works in all case either. Further, the set of things you can rule out with 100% certainty is to small to be useful.
Don't see how that relates to my suggestion of a probabilistic answer though. Has anyone proven that you can't make a statistically valid statement about the "Is a Turing machine" question?
Consider the intuitively simpler problem of "is something a universal turing machine?" Consider further this list of things that are capable of being a universal turing machine;
- Computers.
- Conway's game of life.
- Elementary cellular automata.
- Lots of Nand gates.
Even a sufficiently complex shopping list might qualify. And it's even worse, because knowing that A doesn't have personhood, and that B doesn't have personhood doesn't let us conclude that A+B doesn't have personhood. A single Transistor isn't a computer, but 3510 transistors might be a 6502. If we want to be 100% safe, we have to rule out anything we can't analyze, which means we pretty much have to rule out everything. We might as well make the function always return 1.
OK, as bad as that sounds, it just means we shouldn't work too hard on solving the problem perfectly, because we know we'll never be able to do so in a meaningful way. But perhaps we can solve the problem imperfectly. Spam assassin faces a very similar kind of problem, "how can we tell if a message is spam?" The technique it uses is conceptually simple; Pick a test that some messages pass and some fail. Use the test on a corpus of messages classified as spam and a corpus classified as non-spam, and use the results to assign a probability that a message is spam if it passes the test. In addition to the obvious advantage of "I can see how to do that for a non-person predicate test", such a test could also give a score for "has some person-like properties". Thus we can meaningfully approach the problem of A + B being a person even though A and B aren't by themselves.
What kind of tests can we run? Beats me, but presumably we'll have something before we can make an AI by design.
One problem with this approach is it could be wrong. It might even be very wrong. Also, training the predicate function might be an evil process - that is, training may involve purposely creating things that pass.
Cannon!Snape has loved Lily since the two of them were children - considerably longer than 11 years. I don't think it's unrealistic at all. While I wouldn't call such a love typical human behavior, it's also not particularly rare. There are thousands of people who still profess love for Princess Di for example.
I doubt that it was telling Snape what an idiot he is that angered him, but rather saying Lily was shallow and unworthy.
I agree that it's weird that someone who could carry a torch for that long would stop just because an 11 year old boy gave them random advice. I think it's likely that when Snape kills Dumbledore, it's going to be because of his love for Lily and Dumbledore's interference in that. His love hasn't diminished at all.
The P.S. doesn't grant unlimited wealth, it grants unlimited gold and/or silver. A large part of the value of Gold is related to it's scarcity, so teaching others how to make stones would affect Flamel's personal wealth - oh, and probably destroy society too. And making everyone immortal includes the Voldemorts, the Grinwalds, and Baba Yagas of the world. and it's not like he personally is killing those people...
See how easy it is to rationalize letting everyone die? And I came up with those in just a few minutes - imagine having six centuries to make excuses.
The "All possible worlds" picture doesn't include the case of a marble in both the basket and the box.
They weren't planning on it, but the information was sent nonetheless. P(Someone is going to go back and stop them from going back|They came back) < P(Someone is going to go back and stop them from going back|They did not came back)
That presupposes that P(Bob came back) is not affected by your decision to send the information further on. I'm postulating that IF you would have sent the information further back, THEN P(Bob came back) = 0. Of course, it might not actually work that way, but if my supposition is correct, then Bob not coming back tells you nothing. The event only carries information if you aren't going to make use of that information.
Perhaps the reason he didn't is because you would have sent that information back in time, and so he couldn't.
But every time someone uses a time turner, they send that information into the past. If it didn't block them then, why would it block them now?
Because you would have sent that information back in time. It didn't block them "then" because they weren't going to send the information further back. The effect could be more subtle - instead of preventing you from succeeding, it could prevent you from trying (don't mess with time) or even make you not think of trying.
Another possibility is that information loses "coherence" the further back it travels.
There are ways of fixing that.
No, you can't "fix" it, you can only reduce the effect. If a signal is weak, you can amplify it. But that only works up to a point. And apparently, that point is six hours, even with magical amplification and correction.
I believe there was a book where the world ended because ...
I remember a short story by Larry Niven - Rotating cylinders and the possibility of global causality violation. The short story first appeared in Analog, was reprinted in CONVERGENT SERIES, and it contains the immortal line "I imagine the sun has gone nova". Because the universe protects its cause-and-effect basis with humorless ferocity.
You don't actually know that Bob didn't see the enemy at the pass, you only know that for some reason, Bob didn't come back and tell you. Perhaps the reason he didn't is because you would have sent that information back in time, and so he couldn't.
Another possibility is that information loses "coherence" the further back it travels. (or forward, depending on which side your standing on) Think of it as a signal to noise problem - six hours isn't the limit, it's the limit of what we can correct for with the magic of the time turners. Prophecy seems to defeat the limit, but only by being nearly incomprehensible.
Or maybe it is possible, but insanely dangerous. There are hints that Atlantis was destroyed by something involving the time stream.
If Harry's theory is right, squibs can't be normal genetic descendants (mutation not withstanding) of wizards, but adultery is a very real, very common thing. Cannon does not rule out the possibility, though given that the books were meant to be accessible to children it's not surprising that Rowling doesn't go into detail on the matter.
So after thinking about it some more, I came up with a possible rationale/rationalization why a wizard's death might be needed.
Assume the "script kiddy magic" theory is right - A powerful wizard can be bind complex magic into a simple to execute script, with a key phrase (and/or emotion or gesture). Thus it wasn't some perverse law of the universe that decided "Wingardium Leviosa" is how levitation is activated, but some perverse ancient wizard.
A Horcrux stores an image of you, and the activation sequence is bound to the death of a wizard. It was meant to be an emergency backup script, activated on the death of the wizard. I.e. the ancient who created it was thinking that when a wizard dies, they would automatically be backed up into a Horcrux. This explains where ghosts come from, and why the ghosts we know of were all wizards. Later, someone figured out how to activate the script without dying. Unfortunately, the method they discovered involved killing another wizard.
A backup is limited by the hardware that runs it, so ghosts, which can only barely be said to run, don't seem like real people. They have limited ability to form new memories, so they seem more like chatbots than people (in the MoR universe). A Horcrux is even more limited unless it can get near a brain, but has some "upload" magic associated which means it can possess people under the right circumstances. Harry could be a Horcrux, in the sense that he might contain a backup image of Voldemort, but it can't (normally) run for much the same reason Voldemort can't cast a spell on Harry. That's why the hat didn't sense it.
It certainly looks like Harry is a horcrux in this universe, and Harry already thought of that possibility in different terms, yet the Sorting Hat says with 100% confidence that there is no extra "mind, intelligence, memory, personality, or feelings" in Harry's head.
I would add that in Cannon, Harry is a horcrux, which adds a fair amount of weight to the idea.
Some possibilities for why the hat would make the statement;
- Harry's scar isn't a horcrux.
- A horcrux is nothing like a ghost, mind, intelligence, memory, personality, or feelings.
- The sorting hat was wrong, or lying.
- Something about Harry-the-horcrux prevents detection by sorting hats. For example, it can't be active (and therefore the hat can't detect it) unless Harry is in dark side mode.
- Something about Volemort's horcrux is different than the hat expects. For example, an occlumens can hide from detection.
While I like the idea that the horcrux is only active when Harry is in dark-side mode, I can't see any reason to favor that theory.
In Cannon you had to split your soul, which according to Slughorn required an act of evil. The supreme act of evil - murder.
If Slughorn is right, then no, a willing sacrifice wouldn't do it.
He implies though, that it's not the external consequence of the act that counts, so much as the internal soul wrenching aspects. For some, it might be enough to strangle a puppy. And as you progressed in evil, murder most foul might not be sufficient to tear at your soul. When you've killed four, it's easy to make it five.
I'd much rather get a reply than a vote.
But presumably there's a reason for the current system rather than the arguably simpler method of not having up/down buttons.
The idea is to make it possible to say (by voting) "even though I think you're wrong, I'd like to hear more". The problem IMO with the current system is that the people who vote "I think that's wrong" drown out the people who vote "I think that's interesting". It may be that isn't supposed to happen, but that seems to be what does happen. Would a "rhymes" button make sense? Sure - if you wanted to encourage rhyming posts. The GP wants to encourage contrarians and skeptics, so "like/dislike" and "agree/disagree" seemed appropriate. I haven't seen many of them on LW, but on other boards I really wish there was a "WTF? didn't understand your post" button, as I would press that one quite a bit. What buttons are best is a subject unto itself, but probably not worth discussing unless the basic concept is possible and worthwhile.
I think the kind of people you're looking for are rare in general, so it shouldn't be a surprise that they are rare on LW.
That said, there's room for improvement. The karma system only allows for one kind of vote. It could be more like Slashdot and allow for tagging of the vote, or better yet allow for up/down voting in several different categories. If a comment is IMO well worded, clear, logical, and dead wrong, then it's probably worth reading, but not worth believing. Right now all I can do is vote it up or down. I'd like to be able to vote for clarity and against content at the same time. And as long as I'm wishing, I'd also like to be able to vote just to vote, so we can have user generated polls without needing a karma dump. And humor - that deserves it's own category. Better feedback, better results. Or at least, so I believe, never having had better feedback.
The only cannon example is Voldemort who mangled his soul six or seven times. A single Horcrux might be less destructive. Also, we may be confusing cause and effect. But then we also have no examples of a Horcurx actually extending life - Voldemort's was cut short despite making several.
I would also like to point out that it's possible to value diversity. The utility of a single point of view for 200 years may not be as great as two points of view for 90.
Tom Riddle: "And how exactly does one split his soul?"
Slughorn: "Well, you must understand that the soul is supposed to remain intact and whole. Splitting it is an act of violation, it is against nature."
Tom Riddle: "But how do you do it?"
Slughorn: "By an act of evil -- the supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: he would encase the torn portion --"
MoR!Horcrux might be different, but it seems likely that killing a willing victim isn't good enough, it has to be murder most foul. If a MoR!Horcrux is different, and only requires a death, then why assume a human death is required?
Dumbledore doesn't come right out and say it, but it's there in Chapter 77;
"In other words," the boy said, as though talking to himself without any other people in the room, "it's already known how to keep powerful Dark Wizards in prison, without using Dementors. People know they know that."
Lupin was brought in as a special instructor for the Patronus charm, thus might possibly have professor status.
The evidence against QQ is pretty strong;
- H&C cast a memory charm on Hermione without triggering the wards.
Only a Hogwarts professor, Dumbledore, and maybe Lupin could do that.
- H&C either wanted Hermione to be blamed for attempted murder, or wanted her to succeed.
That rules out everyone is isn't willing to kill innocent children to advance their plots. Realistically, only QQ fits.
Whether Harry has enough information to figure it out is a different question.
I'm worried I may be turning Bad.
You need not trouble yourself. Examining Quirrell's actions has merely made you realize how much you would like to have his power. "Bad" is just a label applied by those too weak to seize that power.
Do not fear the dark side - we have cookies!
So, you think he's not SC but wanted McGonagall and Snape to think he was? In that case, why carefully evade the question rather than just lie?
Some people believe actions carry more moral weight than consequences. To such a person, a lie of omission is a lesser crime than a bald faced one. They might, for example, respond "only a fool would say yea or nay" rather than actually answering a question, or quote some obscure piece of text and hope that you drew the wrong conclusion from it.
I don't see fiat as something the wizarding economy can jump straight to. First they have to be sold on the idea that money is a medium of exchange, and that the ability to exchange it is it's primary value.
Representational money doesn't have to be mono-metallic, it could represent a basket of metals, or a basket of any commodities for that matter.
Certainly, the planning fallacy applies. And even if, for example, arbitrage worked the way it seems, and without the extra pitfalls that have been mentioned, there's a lot more to it than just swapping silver for gold and back. Harry's 11, he can't leave Hogwarts, his finances are tightly controlled by Dumbledore, 100,000 galleons = 1.7 million sickles ~= 17 tonnes of silver. Your dad doesn't just slip that into his back pocket. You're going to need help lifting it, security to guard it, vehicles to move it...
On the other hand, Harry has a lot of resources that haven't even been mentioned yet. There's a house in Godricks hollow for example, and the Granger's would probably be willing to contribute.
He hasn't even really made an accurate count of his vault. He described the stacks as a rough pyramid, but then estimates they're 20 wide and 60 tall - so in other words, each step of the pyramid is only three coins high. I made a small model out of poker chips, and it looks more like a flat than a stack. If it were a normal author, I'd figure the description was bad and the "estimate" was spot on, but EY is smart enough to realize that estimates aren't that accurate. Harry might have underestimated and already have 100,000. Of course, he might have over estimated instead.
Maybe he should learn a magical counting spell.
Doing something stupid, or just being an idiot in general isn't the same as holding the idiot ball.
the person carrying the idiot ball is often acting out of character, misunderstanding something that could be cleared up by asking a single reasonable question or performing a simple problem-solving action, but that he isn't doing solely because the writers don't want him to. It's almost as if the character is being willfully stupid or obtuse.
Mostly - fail.
For a detailed explanation, I recommend "The Big Problem of Small Change" - Thomas J. Sargent, François R. Velde published in 2002 isbn 0-691-02932-6
A common strategy was to ignore the problem and hope it goes away. It didn't - see Gresham's law
- bad money drives out good. Another strategy was to debase the more valuable coins (make them of less pure metals), which has approximately the same effect.
All of which leads to "economic hardships" on the poor, which sounds a lot nicer than "the poor died in droves."
It seems strange, but the idea of a representational currency, coins that you can officially exchange for a fixed amount of gold, is actually a relatively recent invention. Not to be confused with fiat money - coins that you can't officially exchange for anything.
Edit: Did you mean "how did countries set the exchange rate in the old days?" If so, then typically the government reserved the right to mint coins, and the exchange rate was set by law.
He also didn't get Hermione pregnant.
My thinking was more mundane; gold foam with a solid shell. But yeah, seems like there's a lot of possible sources of error in the size/weight of a Galleon magic or no. Still, given the volatility of the Muggle marketplace and the isolation of the wizarding world in general, it seems likely that some arbitrage opportunities exist.
Not that I should care about the destruction of a fictional economy, but I much prefer the idea that arbitrage is only, say, 10%, and Harry decides to strike a long term business relationship with the goblins rather than taking an adversarial position and crashing the market in a big way. He could even introduce the idea of representational currency, which ought to be worth quite a bit to the goblins.
Read it? Probably.
Understand it despite the Interdict of Merlin? Not so sure.
Anyone know if Galleons are solid?
Harry estimated their weight at 5 grams, about 1/10th of what a solid gold coin about 38.6 mm in diameter would weigh.
nod --- I see.
Harry thinks Hermione is innocent, and he's probably deluded enough to think that proving it to the Wizengamot will make a difference to them. He's not likely to give up Dumbledore or someone he cares about permanently when in his mind Hermione's plight is temporary.
It seemed to me that Harry didn't catch on that the call for Azkaban was a set piece, that Lucius must have spent significant political capital to get it to happen the way it did. Nor did he seem to realize the implications of Dumbledore thinking about giving himself up instead of dismissing the idea out of hand, but perhaps I'm wrong.
There was speculation before Chapter 79, but H&C as anyone but a Hogwarts professor is killed by McGonagall's comment;
Obliviation cannot be detected by any known means, but only a Professor could have cast that spell upon a student without alarm from the Hogwarts wards.
There's some minor speculation that an ex-professor could have done it, and I suppose we could include Dumbledore and Lupin in the list, but Sirius and Grindelwald are ruled out (as of chapter 79).
Someone (sorry, I don't remember who) commented that the apparent sloppiness of the Ground Hog Day attack may be from the need to explain to the reader what's going on. It's also been pointed out that being a Legilimens doesn't guarantee competence in gaining trust, it just makes it easier to figure out that you haven't. As is typical, YMMV.
Much hay has been made of the line
and recognition sent a jolt of terrified adrenaline bursting through her.
With some claiming that it means Hermione both recognizes and is terrified of the real H&C. Others claimed that was reading too much into the prose, and the terror might just be from realizing the situation in general. Again, YMMV.
Current speculation seems to focus on the motive, as in, who would want Draco dead, and/or who would want Hermione accused of trying to kill Draco.
It seems likely to me that it will all become moot once chapter 81 is released, so get your speculations in quickly!
I think it's fairly clear that Snape has moved on from Lily now ...
It's not clear to me. What has Snape done or said that makes you think that?
Use a false memory charm on a student to generate testimony framing someone else as false memory charming Draco and Hermoine.
That would fall under 5. "... find someone and give them up as Dracro's assailant/Narcissa's killer, without considering their actual guilt." And like any option that falls under that broad category, we don't know how long it would take to carry out, so it's more "Let Hermione go to Azkaban while framing Lord Jugson." (action 4 plus 5)
If I were going for the safe, boring route, I'd pick 4 combined with trying to determine (and prove) the actually guilty party who false memory charmed Hermione. There aren't very many people who;
- Are Hogwarts professors (i.e. someone who could cast the memory charm without triggering alarm).
- Have a motive.
"Albus," Minerva said, surprised at how steady her own voice was, "did you leave those notes under Mr. Potter's pillow?"
Severus's hand halted an instant before casting Floo powder into the fire.
Dumbledore nodded to her, though the accompanying smile seemed a bit hollow. "You know me far too well, my dear."
Is this supposed to be proof positive that Dumbledore is Santa Claus? A nod, and an empty statement?
Between chapter 80 and 81, here's my analysis. I can think of seven broad possibilities;
1.) Do nothing
2.) Attack publicly
2b.) Attack publicly in disguise
3.) Stealth attack
4.) Retreat and regroup
5.) Change the board
6.) Deus Ex Machina
1.) Do nothing; I list this simply because people often forget that inaction may be the best possible action. Here, that doesn't seem to be the case. On the other hand, once you realize that sacrifice is necessary, why not give in to the dark side? What's one muggleborn more or less? With proper obliviation Harry can literally forget about Hermione. Plus, the dark side has tasty Hufflepuffs. And cookies.
2.) Attack publicly.
While romantic, this puts Harry into a massive, wasteful, battle with basically all of wizarding Britton.
He's good, but realistically, he'd lose.
Blackmailing the council publicly seems equally pointless.
Even if they gave in, it would be disastrous in the long run.
On the plus side, this is by far the most dramatic possibility. It's not hard to imagine Harry laying waste to the Dementors essentially freeing all the prisoners, and throwing the wizarding world into complete and utter chaos.
2b.) Attack publicly in disguise. Basically, put on a mask and break Hermione out of custody. Again, several possible ways to do it, but all with the significant drawback of making Hermione a wanted fugitive.
3.) Stealth attack. Harry and Quirrell almost succeeded in getting Bellatrix out without any help and without anyone knowing. With the order and the aurors attempting it, it wouldn't be unimaginable that they could remove Hermione without anyone finding out. On the minus side, Hermione would have to become a non-entity for 10 years, and they'd have to sneak her back in. On the plus side, the comedic potential is enormous. Almost every major character could reasonably have a motive to sneak Hermione out, even the evil ones. Massive Gambit Pileup ensues.
4.) Retreat and regroup The bad part of this is Hemione will be in Azkaban for some amount of time. The good part is that it doesn't result in Harry being at war with magical Britton before he's ready. This probably isn't as bad as it first seems. Many will be outraged by the decision to send a 12 year old girl to Azkaban, including, I bet, Draco Malfoy. Now that the court room acting is over, "Draco" can argue for leniency, and Lucious can soften his heart over the plight of so young a girl who was clearly unaware of the severity of her crime, and blah blah blah. Lucious wants Hermione away from his son, discredited, and wants to stir up the blood purists. All that is accomplished if Hermione is locked up in, say, Nurmengard, and his son vows to stay away from her. Add in a bad publicity campaign smearing the wizingamot "Malfoy says 12 year old girls should be tortured in Azkaban". Harry might even be allowed to visit and banish a few dementors, rather than having to do it clandestinely
5.) Change the board. Determine who really cast the blood chilling charm. Find out who killed Narcissa Malfoy, and give them up. Or the dark path - find someone and give them up as Dracro's assailant/Narcissa's killer, without considering their actual guilt. (Harry could put his minion Lesath Lestrange to good use.) Find something else that Malfoy wants, like say, the philosopher's stone, and give it to him. Become god. All good things to work on, but their timing is not under control, which means this really is a variation of one of the other options with extra work added.
6.) Deus Ex Machina. The author could make anything happen. While it might be that only the author can save them now, it's not something I'd expect the characters to plan for. And I for one would feel cheated if that was the final solution.
Sure, all the prisoners who have family/friends in the order sufficient to provide 24/7 support, that believe the prisoner is wrongfully imprisoned, and have the support of the Aurors are already being protected. The rest have to make due with the occasional visit and bribe their way past the aurors.
They would of course believe it to be a temporary solution, just until they can commute Hermione's sentence to a lighter/more appropriate one, but as the saying goes; "there's nothing more permanent than a temporary solution."
Is it conceivable that Hermione will spend time in Azkaban without protection from Dementors ...
It's not reasonable that Hermione would be unprotected. Everyone in the order of the phoenix knows how to cast a patronus and send it to someone else, and Harry could do a lot more than just protect her from Dementors if it came to that. Plus the chief auror has already said that the aurors wouldn't stand for a 12 year old being exposed to Azkaban, about the only way I can see Hermione being in Azkaban is with 24/7 patronus guards. Anything else leads to open revolt.
I can't conceive of something being inconceivable.