Equation That Shouldn't Exist

Planck's idea, while very unsatisfying for Lucy, worked great. They followed Kleine through the forest, which was both easy and challenging. Her reddish brown fur was easy to spot in a forest, but following her was hard because of all the trees being so close to each other.

-It's weird. Don't you think those two trees look the same? And those four.

-Now that you mention it, don't trees usually look the same?

What he said had some grain of truth to it. If Lucy thought "make all trees look the same", then they would look the same. Similarly if she thought "trees are actually very small" they would shrink, and everyone would notice nothing weird. She didn't think any of that, but sometimes dreams do weird things. One of many reminders that what she saw wasn't real.

Their tree gazing had to end as they arrived in front of the forbidden cave. It was hard to tell where exactly it was located in the forest, since they took so many turns. It was part of a mountain, its entrance perfectly hexagonal. Big enough to let a dragon comfortably enter and exit. Or for rebels to use given poplar perception of this cave as dangerous.

-Kleine, stay in front of the entrance and watch out for any danger. If someone is coming give us a warning.

Kleine licked Lucy and laid herself down next to the entrance.

-I doubt there's any light in there, take this.

Lucy pulled two burning torches from her bag, one for Planck and for her.

-Give me your hand, we don't wanna get separated in there.

Lucy took the lead and they entered the cave. It was spacious and smelled of ground right after the rain. Many stalagmites and stalactites were ever so close to meeting and about to form a pillar, but it seemed as if there was an ever infinitely small space between them. Water was running down the walls, forming a small, river-like body of water. No vegetation was present.

-Lucy wait.

-What is it?

-I think I stepped on something. Something soft.

-It's a cave, it's only natural there is some guano here and there.

-I don't think it's guano.

The 'something' Planck stepped on wasn't guano. It was much larger, longer and more dangerous. It moved from beneath his foot. Planck followed the scaled tailed with his eyes and froze when they landed on creature's head. It also was scaly, bigger than Kleine's and had a magical rune on its forehead. As it opened its mouth, a loud roar and a complaint filled the cave. Lying in front of them was a dragon. They were assumed extinct hundreds of years ago, but this one must've missed the memo. Big, purple and angry.

-How dare you step on my tail human?

-I'm sorry!

-You better have a good reason to wake me from my slumber. I had such a great dream.

-What did you dream of?

Another quirk of lucid dreams was you could get insights on your unconscious. Imagine getting asked "what do you think dragons dream of?", unless you've thought about it before, it would be pretty hard to answer. Asking a dragon made entirely of your thoughts directly is much simpler.

-Smoked beef.

Right, simplest answers are simplest.

-We're looking for people you have kidnapped!

-People? You're the first people I've seen in years. Do you think I've secluded myself in a cave because I'm a 'people dragon'? I long for peace and quiet you have disturbed. Leave if your life means something to you.

-You spend your life secluded in a cave, sleeping?

-You have a problem with that? - Dragon and Lucy replied in unison.

Planck seems to have struck a chord with both the dragon and Lucy. It was hard to say who was more dangerous. Fire has exited dragon's nostrils and filled the ceiling. It didn't disappear, it stayed there providing a much better view of the cave. Its walls were filled with numbers, equations and mathematical symbols.

-Let me get a better look on you before I decide you've taken your last breath.

Lucy looked around the walls. What she saw reminded her of something, she couldn't exactly pinpoint what until she saw the one equation that made everything make sense: P = NP.

-You're trying to prove P = NP?!

-You aren't some simple lass if you can see that.

-He... She... I'm sorry dragon, what do you go by?

-My name is Epsilon, like the rune I've been marked by.

-Nice to meet you. So Epsilon is doing what?

-It's quite simple actually. P stand for 'polynomial time', think of it as "simple problems". Think about this: let's say I give you a list of 10 numbers, and you can only check one number at a time. How many numbers do you need to check to find the biggest one?

-Well, all of them. If I find the 9th number is 500 it could be the 10th is 501.

-Exactly! And if the list contained 1000 numbers you would still need to check all of them. If the amount of steps needed to finish a task doesn't grow too fast, it's a "simple problem".

-Then NP is a hard problem?

Epsilon decided to take over and explain this part.

-That's right. Imagine a sudoku. You humans still play this game?

-We do.

-Great. Imagine I give you a seemingly solved sudoku puzzle. It's very easy to check if it's correct. The hard part is actually solving it. The grid may be 9 by 9, but it doesn't take 81 steps, it takes many, many more steps to solve it compared to checking if it's solved correctly.

-So NP means a problem's solution is easy to verify, but problem itself is hard to solve?

Epsilon gave a nod. They seem to start liking the intruders. Lucy continued the explanation.

-The question "Is P equal to NP" asks "If we can easily check the answer, can we also easily solve the problem?" Most people assume the answer is no.

-But they're wrong! And I will prove it to them. With age you humans stop believing you can do something grandiose. Start getting comfortable with status quo. Truly pitiful.

I think I like this dragon. Why did I make them extinct again?

Lucy's digressions came to an abrupt hold. She still had a mission she needed to complete.

-Then if you haven't left the cave in so many years where did the rumors come from?

-It would be because of me.

For the first time Lucy felt cold in Arithmia. A shiver went down her spine. She just heard someone that shouldn't be there, someone uncanny and unfitting to her perfect world. She turned around and behind her stood a blonde man, wearing a black shirt under a white coat, golden necklace with number '1' engraved on it, and a face mask.

-Who are you?

-Your worst nightmare. There is something they haven't told you at the meeting. It's only the younger siblings who went disappeared.

Lucy didn't know how to react. That man felt somewhat familiar, yet shouldn't be there. His clothing was weird, he spoke of things he shouldn't know of. He shouldn't exist.

Now I remember why I made them extinct, they're symbol of humans' primal fears.

Dreams were fully dependent on dreamer's mental state. If they were to think of spiders, spiders would appear, and if they're afraid of them, dream would turn into a nightmare. Staring at the embodiment of fear must have been enough to start a similar reaction.

-Just burn away.

It wasn't the first time her dream has derailed. If she was to make the anomaly disappear, everything would return to normal. She was still in full control. Intruder's entire body was set aflame. On the middle part of his golden necklace where '1' was engraved, it was replaced by number '3'.

-My body is made of infinitely many points. Every point is occupied by a flame. Since there are infinitely many points, I can just move each flame to the next point.

As he spoke, flames started disappearing from his right hand, then his arm, until all that was left was a tiny flame in his left palm, which he closed.

-Hilbert's paradox?

-Fitting for the Paradox Prince, don't you think?

It was at this point Lucy noticed Planck wasn't with her, and that Epsilon hasn't moved. Or maybe couldn't.

-If you want to stop the nightmare I have just the thing. Ever wondered when dragon stops being a dragon?

Engraving on his necklace changed to '4', and Epsilon started shedding scales. One, by one, scale by scale, and then claw by claw, tooth by tooth, dragon was being torn apart.

-NO!

Lucy's scream was enough for the world around her to become less and less visible, as if someone was decreasing opacity. Everything turned black.

***

Hanako was in her bed, drenched in sweat, heavily panting.

"...I need to use the toilet"