Disclaimer: Impossible geometry will not stop JK Rowling.

Chapter 80

The photoessay on the Second Task netted twenty galleons—a solid thousand pounds—a tidy sum for the younger boys. Colin and Dennis insisted on sending Hermione half, even though she offered to give them two thirds. They were the ones she was encouraging to make a business out of it, after all, but they wouldn't take no for an answer. Still, a few weeks later, the first ads ran in the Daily Prophet for Creevey Bros. Pictures, with the sensational offer to pull quality photos off an Omniocular recording. With their names on that photoessay, they were sure to get at least a few customers.

As for Hermione, that ten galleon windfall was soon to come in handy when one of her experiments turned expensive. She had her potions kits, but her royalties for those were only a couple galleons per month.

She was still at her metalworking experiments. She had made a simple tin lining for her cloud chamber to block out background radiation, and she'd had limited success at melting down iron and graphite powder with burning magnesium into small quantities of steel. It was still just a proof of concept experiment, though, and she could see some definite shortcomings. The magnesium powder burned too fast and sprayed sparks all over the place, making it really inefficient. The natural solution was to try to contain it somehow, but that was easier said than done with a solid fuel. It was hard to confine it whilst still giving it sufficient airflow. She tried putting two crucibles together in a crude imitation of a double boiler, but it wouldn't burn fast enough, and what did burn started to damage the crucibles.

So she went back to her chemistry book. After reading up on the relevant chemistry, she determined that she needed something that was self-oxidising, like solid rocket fuel. She momentarily considered using actual solid rocket fuel before she realised that, for one, she wouldn't be able to get ammonium perchlorate from the solid in significant amounts, and for another, the expanding gases would blow up her whole experiment. However, after a fair bit of searching, she discovered a mixture that burned very hot, was self-oxidising, and that she could replicate easily. She laughed when she saw it; it was so obvious.

A muggle who saw Hermione at that moment would have easily pegged her for a witch, what with the cackling and all—until they saw that the answer she'd found was thermite. That screamed "mad scientist".

She recruited Michel to help with her experiment, mostly to make such the Fire-Protection Charms were good enough.

"And why do you need Fire-Protection Charms on the crucibles themselves?" Michel asked. "They're supposed to be fire-resistant already."

"I know, but thermite fire isn't like other fire. It's basically impossible to put out, and it can melt through an engine block."

"What's an engine block?"

"A five-hundred-pound block of mostly solid steel."

"Oh…okay, then, I suppose we should protect them," he said nervously.

"I'm not too worried. I'm only using a small amount. I just don't want to destroy the crucibles."

Hermione had cleared away what she thought would be a sufficient area in the potions lab for her experiment and set up her rig. Pre-mixed iron oxide and aluminium powder extracted from the soil filled the lower crucible. A sprinkling of magnesium powder would ignite it. In the upper crucible, she'd already moved on from steel, and found a tougher test: pure chromium. The upper crucible was covered to keep thermite droplets out, and the whole apparatus was suspended above a sink, also covered by Fire-Protection Charms, to minimise damage to the lab. It all looked good on parchment.

"Do you see anything else we need to do?" Hermione asked Michel.

"No. I think you're being much more cautious than necessary."

"Oh…well, you'll see soon enough. Glasses on." They both donned safety glasses, and Hermione pointed her wand: "Incendio."

The magnesium powder ignited. There was a bright light, billowing smoke, and then—

CRACK!

CRASH!

"AHHH!"

"HERMIONE!"

Crack-hiss! Patter patter patter.

Hermione was flat on her back, her face stinging. The glasses had protected her from the worst of it, so she hadn't damaged her vision as far as she could tell, but they didn't cover everything. Michel was crouching over her, frantically trying to put out the flames on her robes and around the area. Water was falling around them.

"Hermione, I'm so sorry! Are you alright?" he asked frantically.

"I think so," she said. "Am I missing…an eyebrow?"

"Er…yes, both of them."

"Lovely. Help me up, please?"

She staggered to her feet with Michel's help and investigated the scene. It wasn't hard to figure out what had happened: the thermite had burnt clean through the Fire-Protection Charms, shattered the crucible, chewed through the metal of the sink, and hit a pressurised water pipe, causing the leak. It also sprayed enough sparks to burn her eyebrows off and cause a few nick-sized burns, but it was nothing L'Infirmarie couldn't fix.

Then, as she looked, a frantic Madame de Cotte ran into the room with wand drawn. "The wards registered an explosion!" she cried. "Are you two alright? Was anyone else here? What were you doing?"

Hermione hung her head: "We're alright, Professor, just a little singed…I'm afraid it was my fault, Madame."

"Mademoiselle Granger!"

"No, it was my fault, Madame," Michel said quickly. "I botched the Fire-Protection Charms. I don't know how I did them that badly—"

"You couldn't have messed up all of them," Hermione insisted. "I think the fire burned through them."

"What?" Madame de Cotte said. "Burned through them? What do you mean?" She approached the scene and cast some diagnostic charms. She gasped when she saw the Fire-Protection Charms really were cast properly and had failed: "Mon Dieu, were you playing with Feudeymon in here?"

"Um…I don't think so," Hermione said. "What's Feudeymon?"

"An extremely dark cursed fire that's almost impossible to control."

"Then no. I was experimenting with a thermite fire. Muggle technology. It's twice as hot as normal fires. I guess the charms couldn't handle it."

"I'm sorry, Madame, I should have realised—" Michel started, but Hermione cut him off.

"No, Michel, this was all my idea. I take full responsibility, Madame. I'll pay for the damages."

"Wait, slow down," their Potions Mistress interrupted. "Mademoiselle Granger, you said this…"

"Thermite."

"Thermite burns twice as hot as an ordinary fire."

Hermione nodded: "It burns up to twenty-five hundred degrees Celsius."

"Twenty-five hundred!" That was just a number to her—but an alarmingly large one. Most purebloods never heard of anything that hot in their normal experience. "What on earth were you thinking doing an experiment like that indoors?"

"I thought the Potions lab was designed for it, and the Fire-Protection Charms would be enough, Madame. I've never heard of them failing against a non-magical fire."

Madame de Cotte probably wanted to scold her more, but the truth was that she had probably never heard of such a thing either. "And just what is this 'thermite'?" she pressed.

"It's a simple mixture of powdered iron oxide and aluminium, although it takes a lot of heat to ignite. Muggles usually use magnesium."

"Iron oxide and aluminium did…this?" she shook her head. "Madamoiselle Granger, I've been concerned about your 'experiments' for some time. You've conducted them safely up until now, and you appear to have been taking standard precautions in this case, so I will give you the benefit of the doubt for now, but I forbid you from experimenting with thermite or anything similar indoors or without my supervision. And I will expect you to pay for the damages to this sink. Is that understood?"

"Oui, Madame."

"Good, now I want you both checked over in L'Infirmarie."

Hermione actually felt relieved after that. She was sure she would get a detention for that debacle. She didn't know Fred and George did it. They had getting in as much trouble as possible without being expelled down to an art form. She didn't think her nerves could take it.

Michel, meanwhile, looked a little afraid of her. "Hermione…" he said, "I had no idea you could harness that much power with chemicals."

"It's not really that much," she countered. "True, I had to reduce the aluminium to its pure form, but in these quantities, there's a lot more energy in a Blasting Curse, and that's not even accounting for the fact that magic doesn't conserve mass half the time, let alone energy. It's all in how you use the energy that matters. Concentrate a fairly small amount of energy into such an intense heat source, and it starts doing strange things…like burning through Fire-Protection Charms. You know, dos moi pa sto, kai tan gan kinaso."

"Wow," he said, impressed, "I'm not sure I want to see what would happen if you tried to harness a lot of power."

Hermione shrugged. That wasn't the sort of thing she paid much mind, since she hadn't got into large amounts of power much. Although if she managed to burn her eyebrows off with this little experiment, she should really pay more attention.

It took over a week for Hermione to convince Madame de Cotte to let her try the experiment again. Even then, she was only allowed to do so outside the castle, on a rocky patch of ground with no flammable materials, with strong Fire-Protection Spells over the whole area, at a safe distance of at least ten metres, and with Madame de Cotte personally supervising. She also had to pay a galleon in advance for two graphite crucibles that she hoped would be strong enough to withstand the thermite.

Once she finally looked over all of the safety measures and let Hermione go ahead with the experiment, it went off without a hitch. The graphite crucibles held up to the heat, and the safe distance from the splatters was a good deal closer than ten metres, though it still spewed out frightening amounts of sparks of molten iron than ate into the stones around the experiment. But for all her trouble, she successfully produced a solid medallion of chromium.

However, once she got to that point, she hit a dead end. She had been nursing an ambition to collect samples of all of the (non-radioactive) elements assayed from the soil for a while. After all, what practical application would this research have besides putting a collection like that together? But she wanted all of those elements in solid form and not powder—for convenience—and she couldn't figure out how to work up to the most refractory metals. Thermite would melt almost anything, but not a few of the rarer metals and most notably tungsten, which had the highest melting point of any element. Tungsten was so tough that it wouldn't even melt under an oxyacetylene torch, and Hermione wasn't convinced that the graphite crucibles would hold up to that.

So she decided to change course a bit. She would start working on filtering spells for the rest of the elements—something that would take months given the small amount of time she had to spend on it. Ideally, once she had all of them, she was thinking about digging up one metric tonne of soil and separating out all the metals to measure their concentrations for herself, but she would have to worry about more pressing concerns first—like the Third Task.

With magical help, Hermione's eyebrows had almost finished growing back. Michel, however, never quite warmed back up to her, although he remained a good study partner. She was afraid she had scared him off, but there wasn't much she could do about that. She was going to keep working on her own terms, and she definitely wasn't going to slow down just for a boy…at least not until after the Tournament was over.

She definitely appreciated Michel's help in learning experimental spellcrafting. Though a minority of the advanced curriculum, it gave her much more trouble than the mathematical parts. One of the most advanced lessons of the year was dismantling and reassembling an Expelliarmus in mid-flight.

"Freezing a spell in midair is pretty easy if you use a ward," Michel explained. "With most wards, the spell will splash off because it hits too hard. That's better for most things, since you don't want hostile energy to build up in the system, but not for this. To catch a spell, you need several weak, fast-charging wards spaced a few centimetres apart to slow it and contain it without disrupting the energies."

"Like a backstop at a firing range?" Hermione asked.

"A what?"

"A loose pile of dirt to slow muggle bullets without having them bounce off?"

"Bullets? Oh, from those metal wands muggles use to kill each other."

How do wizards ever blend in with the muggle world? Hermione thought, but she didn't say it. She vaguely remembered the British Ministry using very similar words about Sirius when he was wanted in the muggle world. "Er…yes," she said. "I saw that method in the course book, but is that the only way? I think I've seen references to people catching spells on the ends of their wands, too."

"Ends of their wands?" Michel said in surprise. "Certainly not for this kind of work. I've heard of it, but I think it's more of a charms master's party trick. It's very difficult and takes a lot of precision. You'd only want to do it under controlled conditions."

"Oh, of course…Still, it might be an interesting trick to learn, if you know where to find out more."

"Uh, well, I can look, but I think we should get to work here."

"Right, right," she said.

They set up the Spell-Freezing Ward in a door frame. Two long, wooden blocks with runes carved into them at intervals defined the individual layers of the ward, and they were set on the floor at each side of the door frame. It was a crude, cumbersome way of doing it, but it was simple, with fewer dependencies and abstractions to worry about, like programming in BASIC instead of C++. Both of them knew enough runes that they could have written out the whole ward scheme on one small block, but this was quicker and more reliable, and it was easier to see what they were doing. Once the ward went up, Michel cast an Expelliarmus at it, and the spell froze in the air on contact like a fly trapped in honey.

"Now, that is pretty neat," Hermione said. Never mind that the ward wouldn't hold up to anything more than a Stunner. It just looked more impressive than a regular shield. "Partieséparé," she cast, and the individual components of the bolt of red light—a half dozen magical fields that fit together like parts of a machine—stretched and separated from each other as if held apart by an electric charge. With this view of the spell, she could see visually how the arithmantic elements translated to real magic and came together to achieve the desired effects.

"Good," Michel said. "Now, the tricky part is that some of these magical fields are holding the rest together. Pull them apart in the wrong order, and the whole spell will fly apart. So, if I do this…" He pulled off the outer cylinder of the spell—what looked like the most obvious first move, and the rest of the components exploded out of the wards in a shower of sparks.

Hermione squeaked loudly in surprise. She was starting to understand why experimental spellcrafting was considered so dangerous. Those sparks wouldn't give her so much as a sunburn, but working with more powerful spells, one wrong move could mean disaster. Assembling the spell from scratch, which they did after they understood how to disassemble it, was even worse. You had to conjure the individual magical fields in the air correctly and then put them together. Do it wrong, and the spell could blow up in your face the moment you pulled it out of the ward. It wasn't hard to understand what had happened to Luna's mother. Hermione didn't think she would ever be as comfortable with these experiments as with the good, old-fashioned maths.

Dear Hermione,

I've finished the latest batch of tests you wanted from me. I've enclosed samples of saltpetre transfigured with the various schemes you've proposed along with the control sample of natural saltpetre. Mind you, this would be easier if you didn't need such large samples. Isn't there any way to improve that detector thing you have?

In any case, I've also enclosed several pages of notes on the diagnostics and untransfiguration potentials of the transfigured samples. However, I'm not convinced you're doing an exhaustive search. Can you explain in clearer terms how you can prove arithmantically that there is no class of non-alchemy methods that allows transfiguration of radioactive particles, based only on the tests we've done?

Sincerely,

Rebecca

The beauty of partial differential equations was that it made any kind of spellcrafting involving waves amazingly simple—well, simple wasn't exactly the right term, but it gave some very straightforward ways to describe them. Using wave equations in spells was an advanced technique already known to arithmancers, but Hermione was able to take it a step further, extending her experiments throughout the electromagnetic spectrum, and she had a few ideas for sound waves, as well.

The first thing she noticed was that the electromagnetic spectrum was not well explored by wizards—unsurprising, since even muggle scientists didn't know that light other than visible light existed until 1800, and its extension beyond ultraviolet and infrared was not proved until 1886. So wizards had never bothered much with anything but visible light.

The second thing she noticed was that, when she tried to adjust her Laser Charm to shorter and shorter wavelengths, the spell unexpectedly became much harder to cast, and that very suddenly. The efficiency of the spell dropped by well over half at a wavelength of—she measured it carefully—exactly 285.6 nanometres. It took her a while to figure out just why such a sudden change should occur. Eventually, she found the appropriate tables, did the maths, and found that that wavelength corresponded with the ionisation energy of potassium. Evidently, magic didn't cooperate well when things started getting ionised. And perhaps it made sense: ionising radiation was at the root of what she was trying to disprove with Gamp's Law.

"Holy cricket! That could be worth a paper in Challenges in Charming all by itself," she said to herself, "not to mention being a valuable step towards my eventually proof against antimatter. Oh, yes, this is good."

To test her hypothesis that ionisation was to blame, she pushed more power through her wand and shortened the wavelength further. Sure enough, the resistance suddenly jumped a lot more at a wavelength of 241.2 nanometres, the ionisation energy of sodium. Potassium and sodium had the lowest ionisation energies of any common elements in the environment. From there, the Laser Charm grew gradually harder to cast as it started ionising trace elements. After she started ionising aluminium, calcium, and magnesium, she could barely keep it going, and she finally hit a wall at 119.7 nanometres, the ionisation energy of sulphur, one of the six main elements of organic matter. At that point, she decided to stop rather than risk collapsing from exhaustion.

All in all, it was probably a good thing. If she could cast x-rays efficiently, giving someone fatal radiation poisoning would be scarily easy.

So she turned around and went to lower frequencies. Infrared lasers were great for burning things and had the benefit of being invisible. After that, she didn't find much, except to probe the effect of magic on electromagnetic radiation. She knew that electronic devices, including radios, didn't work at Hogwarts or Beauxbatons unless they were enchanted to do so, like the Wizarding Wireless. Actual electricity, which ran at much lower frequencies, was blocked entirely, except for natural electricity—brain waves, static electricity, and lightning—those sorts of things. That came back to how many forms of magic had less effect on living things or on nature.

Hermione couldn't really measure the efficiency of her spells quantitatively, but she did notice that casting electromagnetic waves became harder as she lowered the frequency. She hadn't mentioned to Harry that her Microwave Charm was also very difficult to cast inside the wards in addition to not being powerful enough to be useful. After thumbing through an electronics textbook, she had a basic idea of what was happening. As far as she could tell, magic seemed to act as a high pass filter with a breakpoint of about 10 gigahertz, or 3 centimetres, where it was about half as efficient as normal. From there, the efficiency fell by a factor of four each time the frequency was halved. Why the breakpoint should occur there, she didn't know, but it meant that non-magical radio signals were completely blocked and all electronics stopped.

"Hmm…by that logic, though, a high-enough frequency wireless signal ought to be able to punch through the magical field," she mused. But then, she realised the problem: "No, no, no, the receiver would still need electricity to work. Never mind."

The one hard part, the one that gave her a few uneasy nights in the end, was how to test these spells. These weren't like the hexes she had been creating, where they were designed based on what she wanted them to do. These were spells where she didn't know what they would do to living things, and she didn't feel comfortable giving any of them to Harry until she knew what they did. Hermione discussed this concern in rather vague terms with Madame de Cotte and was a little dismayed to learn the wizards were considerably more cavalier about animal testing than muggles—at least with a few select species, like toad, rats, and various insects. Madame de Cotte was much more concerned that Hermione was crafting spells that needed animal testing.

"If you are creating spells that dangerous at your age, I am concerned about you mental state and the safety of the other students, especially after the results of your other experiments."

"I don't think they're really that dangerous, Madame," Hermione said quickly. "Certainly no worse than an Incendio spell. I can even estimate what they do. I just want to be sure before I suggest them to my friends. It's only really for the Tournament, anyway. If it weren't for that, I would file them for future reference and not bother with anything destructive." Okay, that last part was a lie, but she would certainly be more cautious about the whole thing and wouldn't focus on it as much.

Needless to say, Madame de Cotte asked her a number of pointed questions about just what these spells were and what she expected them to do, and only when Hermione had explained them to her satisfaction did she relent and accept the need for testing, directing her to the Care of Magical Creatures teacher to obtain a couple of rats to test on (with supervision, of course).

This was actually scary to Hermione, but in a very different way from most of the scary things she had experienced. She almost didn't work up the nerve to do it at all. The idea of animal testing just galled her, no matter how necessary it was in the muggle world. It's for Harry, she told herself. It's for Harry. She went ahead with it, but she felt like she needed a shower afterwards.

Concentrated infrared radiation burned on contact, as she expected. She was careful to keep the spells low-powered so she wouldn't burn the rats badly. The weird thing happened when she lowered the frequency. Around 95 gigahertz, the rats went nuts, running around like they were on fire, and yet, they showed no visible burns. Diagnostic charms showed they were in distress, but not injured, even with prolonged exposure—but Hermione couldn't take seeing much of that. They looked like they were in a lot of pain, but she didn't see how. She even mentally double-checked her math to make sure she hadn't accidentally discovered a dark curse.

"I don't know, Mademoiselle Granger," her teacher said. "The diagnostic spells say it's perfectly safe, at least for the amount of time you cast it. If you wish to test it on me, I could at least tell you what it feels like."

"I'm…not sure I want to use you as a test subject, Monsieur," Hermione said uneasily. "If it's as safe as you say, I should be able to use it on myself." He started to object, but she said,"I wouldn't want to test anything on you that I'm not willing to test on myself." He reluctantly relented, and she rolled up her left sleeve and steeled herself to cast the spell on her arm.

It was the longest spell she had ever fully crafted—twenty-five syllables. Most of these electromagnetic spells were like that. If she found something useful, she would create a shorter version. She waved her wand very carefully and said, "Repercussiones viribus prope frequentia quae moleculis aqua concusant—YEOWCH!" She dropped her wand and clutched at her bare arm.

"What is it?" her teacher said.

Hermione looked at her arm carefully. There was no sign of any burns. In fact, it felt perfectly fine the moment she let go of her wand. "I don't know, Monsieur," she said. "It felt like I brushed against a hot stove. I don't know why it did that, though…unless it somehow microwave-heated just the very top layer of my skin. Hmm…penetration depths—maybe…But I'm not sure if that one's so useful, either. I might get a reputation as a dark lady throwing a spell like that around."

She decided to keep that spell in reserve. However, that wasn't the worst part. Not even close. The worst of it was the ultraviolet spells. Exposure to intense ultraviolet light gave the rats a bad sunburn. That was to be expected. What she didn't expect, though, not having read up on the details of the medical science, was when they also started having vision problems…two days later.

All the worst curses in the wizarding world worked immediately. Delayed-onset damage? That was scary on a much more visceral level. You'd never know if the curse was done working. Even though the damage was easily fixed, she felt sick when she saw what her spells had done to the rats.

"That's it," she told herself. "No more animal testing unless it's an emergency."

Dear Rebecca,

Thank you for the samples. They were just what I needed. The paper should be ready to publish before the end of the school year. I've enclosed a partial draft including most of our results so far. Please add your parts as you see fit.

The proof comes in three parts. The first part is a set of sub-proofs that demonstrates how each of the transfiguration schemes we've tested can be selected from a broader class without loss of generality. The second part is to show that the general radioactivity terms we've isolated are forbidden as components of each of the classes. (Those individual proofs are pretty simple.) The third part is a proof by contradiction showing that any potential transfiguration spell that might allow radioactivity must fall into one of those groups. I'm still working on the third part, but I fully expect to be able to complete it with a similar argument to the other five exceptions to Gamp's Law. In fact, it should be easier because it's more precisely defined. You can see the parts I've completed for yourself in the manuscript.

Sincerely,

Hermione

The use of human blood in magic was generally considered dark, from what Hermione could tell. (There were many advantages to being an Advanced Arithmancy student, and one of the lesser-known ones was that it was relatively easy to get access to the Restricted Section of the library.) However, for the most part, it seemed that blood by itself meant dark as in distasteful rather than dark as in dangerous, much less dark as in, for lack of a better word, corrupting. True, potions that used human blood usually did nasty things, either as the intended effect or a side effect, but they didn't do nasty things to the donor, or even to the brewer.

This was in stark contrast to blood rituals, which could and very much did do nasty things to the "donor"—things like dark bindings and blood-bound curses and other horrors that Hermione had no desire to invesitgate further. Mixing blood and runes was a little ambiguous because rituals tended to be heavily runic-based, but between her reading and a few discreet questions to teachers and seventh-year students, she was able to determine that all those dark side effects were powered by corresponding runes—runes for binding and cursing and the like.

The upshot was that the runes used to make wands would have no effect on the donor when mixed with human blood. The only bad side effect—and it was only bad depending on one's point of view—was that it would bind the wand to the donor so that no one else could use it. This was an unpopular technique more for cultural reasons than practical ones, although the relative weakness of the resulting wand, even when made by a professional, was an obvious problem.

Hermione wasn't worried about that, though. This was about equal access to magic for muggle-borns. Okay, her motive was more selfish than that. She was just hoping for a simple way to beat the Trace so she could cast magic over the summer—partly out of resentment that purebloods like Draco Malfoy could, and partly out of a single-minded desire not to have to interrupt her studies.

Hermione's studies of wandlore had led her to the conclusion that animal fibres would almost always be superior to plant fibres for wands, and she was pretty sure that this was the last piece of the puzzle she needed to make a backup wand that she could use all summer, without getting too far into the weeds with regards to runes. This experiment wouldn't be a final test, but it would be a proof of concept. She made a "toy" wand, not bothering with the runes, by placing one of her own hairs between the two pieces of wood and adding a few drops of blood to the adhesive.

She gave the completed wand a wave. It felt warm in her hand, like her real wand. And it shot out a shower of sparks. But they had changed some. When she'd first bought her wand from Ollivander's, it had produced sparks that were almost pure white. With this wand, the sparks were tinged blue-silver with streaks of blood red. She wondered if that meant anything, like a personality test. Perhaps, although more likely it would be about as relevant as a horoscope.

Dear Hermione,

I have a lot of stuff to tell you. A lot of it's about the Third Task, but there's a lot more. They called all of the champions out to the Quidditch pitch on Wednesday to see what they were setting up. Professor Flitwick was leading a team of wizard and goblins building something. Bill Weasley was there, too. Some of the goblins started yelling at Bagman about something, but Flitwick told them to shut up and get back to work. I've never seen anyone talk to goblins like that. I think it's because he's part goblin. I sure wouldn't want to try it.

Anyway, Bagman said the Third Task would be some kind of maze filled with traps and creatures and stuff like in old tombs. The Triwizard Cup would be in the middle, and the first person to grab it would win. You get more of a head start if you have more points. The thing is, the maze looked like the start of some weird square building, but it didn't look right—like the parts didn't fit together. I couldn't figure out how to describe it, so I had Colin take a picture.

That's not the weirdest part, though. We heard the full story from Hagrid the next day. He was tending the Abraxans from Beauxbatons when Mr. Crouch came stumbling out of the trees acting crazy and rambling something about his son—the dead one, remember? And then he demanded to see Dumbledore. Hagrid was about to take him up to the castle, but something spooked the Abraxans, and they nearly trampled him, and when he looked again, Crouch was gone. Even Moody couldn't find him, but they found his hat, so they know he was there. No one knows what's going on. Moody thinks maybe he was kidnapped.

But there was another thing. Don't yell, okay? Yesterday, I fell asleep in Arithmancy class. I said quit yelling. I don't know how it happened, but the next thing I knew, I was having another vision of Voldemort—just like back in August. There was a woman there who looked really out of it, like she was Imperiused or something, and Voldemort was talking to someone I couldn't see about someone being dead, and then he said he was going to feed me to his pet snake. That was when Professor Vector woke me up. My scar was hurting, so she sent me to the Hospital Wing, but I went to Dumbledore instead because Sirius said to if I had another vision. Dumbledore said I probably get visions when Voldemort feels strong emotions—something to do with when he tried to kill me.

Actually, Dumbledore said a lot of weird things. He thinks Voldemort's getting stronger somehow. He was using something called a Pensieve that lets you view memories, and he was looking at the trials of some Death Eaters. And he said Snape was a Death Eater, but he became a spy. He wouldn't say how he knew for sure that he'd turned, though.

Also, Fred and George were talking about blackmailing somebody. Do you know anything about that?

Harry

Well, that was disturbing, to say the least. Crouch was acting strange and was possibly kidnapped? Could that have anything to do with the plot that got Harry entered into the Tournament? It didn't make sense. Voldemort was getting stronger, which was very, very bad news. Could he have been involved with Crouch? Probably. With Harry getting into the Tournament? But why? What could Voldemort possibly have to gain by Harry being in the Tournament, win or lose? To kill him? There had to be an easier way. Why not just have whoever entered Harry shoot him in the back with the Killing Curse?

No, she couldn't imagine how to tie it all together with one evil plot—not unless Voldemort had descended to the Bond villain level of foolishness where he would go out of his way to arrange a face-to-face meeting. She had only met Voldemort once—she shuddered as she remembered the Chamber of Secrets—but she didn't think even he was that narcissistic.

And Snape was a Death Eater who became a spy? That was…believable, although Hermione was surprised someone as closed-off as he could give Dumbledore reason to trust him implicitly, but that was a small twist to her in comparison.

Fred and George were blackmailing somebody? She dearly hoped Harry was mistaken about that.

And finally, there was Colin's photo. She examined it with a magnifying glass. It didn't look too odd. Just a large, stone building under construction—perhaps with an inordinate amount of stairs. The only really odd thing was how that one staircase ended at a solid wall…and how that one window was sideways…and that one goblin in the picture came to a staircase and mounted it on the vertical faces…

"Merlin's beard! It's an M. C. Escher print!" she cried.

This was going to be harder than she thought.

Dear Fred and George

I heard a disurbing rumour from Harry…Actually, I heard a lot of disturbing rumours from Harry, but this one concerns you. He claimed to have overheard you talking about blackmailing someone, whom I assume is Bagman. I really hope he misheard. If you're getting into blackmail, I don't want anything to do with it, and I would hope you aren't foolish enough to do anything illegal to get your money back.

Love from,

Hermione

Dear Hermione,

Don't worry, we didn't blackmail Bagman. Harry didn't hear the whole thing. We did send Bagman a final warning letter that we wanted to settle our accounts at the Third Task and at least get our money back, or else. Fred wanted to say or else we'll tell Rita Skeeter, but George said that was blackmail, and we shouldn't do it. We made sure we did everything legal. Anyway, we're going to try to settle things then.

Forge and Gred

"I still don't see why you want to learn this trick," Michel said.

"Just for thoroughness. Besides, it'll look cool if I can pull it off," Hermione replied. And if I ever get a chance to use it on Malfoy, it might just scare his pants off, she added mentally.

After some searching, Michel had found the technique to catch a flying spell on the end of one's wand. It was very difficult, not so much in complexity, but in that it required pinpoint aim and lightning-fast reflexes. It also wouldn't work for a lot of dark curses, the same as a Shield Charm wouldn't. But it was a cool trick, and Hermione wanted to add it to her repertoire. Unfortunately, that meant going through the fairly unpleasant process of having someone cast spells at her until she mastered it.

"Alright, then," Michel said reluctantly. "Are you ready?" Hermione nodded. "Expelliarmus."

"Attrahe!"

Hermione's wand flew out of her hand and into Michel's. He tossed it back to her.

"Expelliarmus."

"Attrahe!"

Same result.

"Expelliarmus."

"Attrahe!"

Hermione tripped and fell on her arse. That one hit a little too hard. Her wand was once again lost.

"Are you okay?" Michel asked.

"I'm fine. Let's try it again."

"Are you sure?"

"I can take it. Go ahead."

Michel sighed and started again. "Expelliarmus."

"Attrahe!"

"Expelliarmus."

"Attrahe!"

"Expelliarmus."

"Attrahe!"

ZZZIP!

Hermione goggled at her wand. Michel's spell was caught on the end of her wand, spinning in place, waiting to be released.

"You did it!" he said in shock.

"I did it!" Hermione said, equally shocked.

"You did it! That was brilliant. Honestly, I didn't think…"

Hermione smirked at him and whipped her arm around like a cricket bowler with a whispered "Libera", and flung his own spell back at him. His eyes widened as it hit his chest, and his wand flew into her hand.

"Okay, I deserved that," he admitted. "I should have learnt by now not to underestimate you."

Dear Rebecca and Septima

I did it! I completed the proof! At least I think I did. I couldn't find any mistakes, but I need you both to check over my maths. But it's all pretty straightforward if you're familiar with the other exceptions to Gamp's Law. The really novel part is the arithmantic construction for radioactivity. I wrote the whole thing up in the draft. If there aren't any mistakes, we should be able to submit before I come back to Britain for publication on 1 July.

Love from,

Hermione

While Hermione was basking in the warmth of having made the greatest arithmantic discovery of the decade—and the Pyrenean sun—she was not idle. Though she had completed a couple of them, one of her projects was still ongoing: her attempt at replicating the magic of the Marauder's Map.

Oh, and she was also perusing the long-awaited proof of Fermat's Last Theorem. How could she not? After all, it had taken 358 years to prove. Unfortunately, it was a hundred-page monster of a proof involving a lot of complex analysis that she hadn't learnt yet. It certainly couldn't be the "truly marvellous proof" that Fermat had claimed. Actually, there was good reason to think that Fermat didn't have a proof, or thought he had one, but it turned out to be wrong, since he worked on sub-cases of the proof later in his life. In fact, Hermione could very easily construct a simple, but wrong proof in just a few lines using ring theory, and it was her personal belief that it was the same proof Fermat had intended before he realised his mistake.

But Andrew Wiles's great discovery aside, she was keeping up with her magical cartography project. She hadn't bothered trying to carve the runes for the Homonculous Charm into the anchor stones of Beauxbatons Castle. She wanted to wait until she had a workable system before risking something like that. She did, however, try setting up a rune stone in her dorm tied into the wards of the school to tell her who was in it. It was tricky, not going through the anchor stones, but it seemed to work. Her small map now displayed people whose names she hadn't individually added to the rune structure.

When it came down to it, there was one big hang-up in making the map. The Homonculous Charm was only really meant to track humans. In its native form, it wouldn't track changes in the shape of the castle, nor did it track non-humans. That meant she had to dig into the mechanics and modify the spell for each species she wanted to include. It took a lot of work just to get Dobby to show up, and that was with him standing right there so she could scan his magical signature for reference. The original Marauder's Map didn't even show elves, but it did show ghosts, Peeves the Poltergeist, and Mrs. Norris, as well as people under an invisibility cloak or transformed as animagi, both of which would defy the Homonculous Charm. (It was an area tracking charm showing their movement through the mapped area and so could be thrown off by simple concealment methods, in contrast to tracking charms on a person directly.)

Come to think of it, there was another problem in getting the map to show passwords to the secret passages. That would require "hacking" the wards a lot deeper.

The other issue was how to display the map in an easy-to-use form. The original Marauder's map was barely readable, with minuscule dots labelled with minuscule names because the only way to fit an entire floor of the castle onto one spread was to make everything really tiny. It would be a lot easier if the map had a zoom function, so Hermione wrote one into the hidden rune layer, controlled by wand taps on a small symbol.

The original Marauder's Map was twenty-two pages long (though it magically folded up as if it were a single page), since each floor of the castle had to be on a different spread. It would be a lot less cumbersome if there was a way to switch between floors on a single page, so Hermione added one, and a scroll function to move around the image.

The original Marauder's Map showed everyone at once. During mealtimes, the Great Hall was an unreadable mess, and even at the best of times, it was a pain to find a particular person amid all the names. So Hermione added a way to show or hide certain names or classes of names. She wanted to add a search function, too, but that proved to be more complicated.

Everyone on the Marauder's Map showed up as an identical black dot. It would be clearer if certain types of people like teachers or ghosts showed up in a different colour, so Hermione added that, too.

It didn't make a lick of sense, how she could do all that. It was so easy, she felt like she had to be missing something. If it were this simple to add all these new functions, why hadn't the Marauders done it? But then it hit her: the Marauders had made their map before the invention of the graphical user interface.

Oh, she was going to knock their socks off.

Dear Hermione

We've looked over your maths, and we couldn't find any mistakes besides a few misprints that didn't affect the proof. This is some of the most brilliant arithmancy I've ever seen—S. We've polished up the draft with our parts, and if you approve, you should be free to submit it to Annals of Arithmancy. We also agree that a letter to Transfiguration Today is in order, summarising the argument and pointing to the A of A paper for the details. That's only fair, and it will get more attention and press coverage, anyway. I recommend sending both papers to TT—S. Professor McGonagall should be very surprised. Please note that if TT picks up our letter in a timely fashion (which they probably will for something this big), it will probably be published while you are at Hogwarts for the Third Task. If you think that will be a problem, we can hold off until next month.

Sincerely,

Septima and Rebecca

Well, that was interesting, alright—quite a coincidence with the timing.

Hermione had openly told her parents when she going back to Hogwarts this time, since she could use the Arithmancy N.E.W.T. as an excuse. It was one less thing to cover up with them. The French N.M.A. exam was on the Tuesday before the Third Task, while the N.E.W.T. was on Thursday. She had arranged to take a couple of her other exams early so that she could go to Hogwarts on Wednesday and settle in. Since the year had gone safely for her (at both schools), her parents agreed that she could come home on the Hogwarts Express the following Monday.

Transfiguration Today would be delivered on Friday, the day before the third task. She didn't have a problem with that. She enjoyed the press attention when it was properly earned and accurate, as opposed to getting caught up in mortal peril against her will, and she knew Harry felt the same way, or he wouldn't be a Quidditch player. So it might be good to get that out there. And if they were lucky, it might even throw Rita Skeeter off her game. A competing story was a competing story, not matter how much she was bugging people.

Wait a minute…

Partieséparé: stylised from the French for "separate parts".

Repercussiones viribus prope frequentia quae moleculis aqua concusant: approximately Latin for "vibrations of energy near the frequency at which water molecules collide". This is a preliminary form of the spell allowing Hermione to swap out different frequencies quickly, and it is longer than usual because there are few natural sources of those frequencies, making it hard to describe.

Attrahe: Latin for "attract".

Libera: Latin for "free".

Hermione's scholarly papers to date:

A paper in The Practical Potioneer on brewing of potions by non-magicals (sole author).

A letter in Magizoology Monthly on blocking the gaze of a basilisk with a blue filter (sole author).

A paper in Annals of Arithmancy about the basic form of her Laser Pointer Charm (sole author).

A paper in Annals of Arithmancy on the analysis of Extension Charms with non-Euclidean geometry (with Septima).

A paper in Annals of Arithmancy on complex structures in Colour-Change Charms based on group theory (sole author).

A letter in Transfiguration Today on the Sixth Exception to Gamp's Law (with Rebecca and Septima).

A paper in Annals of Arithmancy demonstrating the proof of the Sixth Exception to Gamp's Law (with Rebecca and Septima).