It was, Daiyousei considered, a beautiful day on Misty Lake. The sun was shining, making the fog that gave the lake its name glitter like a wall of miniature rainbows, and the frogs were croaking on-key.

"Hey, Dai!"

Cirno's yell was a little off-key, but then, she wasn't a frog. The other fairy came rushing up, her icicle wings fluttering as she hovered alongside her friend.

"You're pretty smart, right?"

"Umm..." Daiyousei was actually fairly intelligent for a fairy...intelligent enough to recognize the fact that for a fairy was what humans called "damning with faint praise."

"So, I was just over near the Scarlet Devil Mansion and I heard the purple lady...you know, the one with the patches?"

"I think that's actually her name?"

"Anyway, she was talking to the sleeping dragon lady, and she asked her a math problem, and I don't know the answer."

Daiyousei blinked.

"Could you start from the beginning?"

Cirno sighed, but she was magnanimous to friends with lesser understanding, as was the righteous obligation of the strongest.

"The purple lady asked the dragon lady why she could never keep that black-white magician girl...I think it's the one I beat up all the time?...from stealing her books. And the green dragon lady got mad and asked why she kept blaming her when she was just the gate guard. So the purple lady said that trying to make excuses was as useful as asking how many angels could dance on the head of a pin. And I don't know!"

"Know what?"

"How many angels can dance on the head of a pin!"

"Is that what she was really asking?"

"I don't care about their silly argument, Dai. I just want to know what the answer to the math problem is!"

"Well, I definitely don't know."

Cirno rubbed her chin.

"That's good, I guess. At least there's no fairy better than I am at math! But still, what if one of the others finds out? Then I won't be the strongest, and that's not right, because I am, and—"

"You should ask someone," Daiyousei cut in before her head got too spinny trying to follow her friend's line of logic.

"But if nobody knows, what's the point in asking them?"

"Not a fairy. I mean a human or a youkai. It doesn't matter if they know, right, because you'll still be the only fairy to know."

"Except you," Cirno said, narrowing her gaze. But then her expression cleared. "But that's okay, because you're just my sidekick on this quest! So who should we ask?"

"How about Alice? She's a magician, so she's pretty smart."

"Yeah! But the Forest of Magic is pretty far away..."

"She's right over there." Daiyousei pointed towards the shore. "She's probably going to visit Patchouli at the mansion."

"Perfect! Even destiny knows better than to get in my way!"

Cirno immediately zipped off towards the shore, Daiyousei tagging along. She wasn't really interested in the answer, but going on a quest sounded kind of fun, and Cirno's games were usually pretty entertaining when they didn't end up in her getting shot at.

Alice Margatroid was a youkai magician who specialized in dolls that she animated and controlled. She had a lot of different varieties, but the two that were hovering alongside her were fairly generic types, with red bows in their long blonde hair, blue dresses, white aprons, and red bows. They probably could charge with lances, shoot lasers, or just explode, so Daiyousei hoped that Cirno wasn't going to be too belligerent.

"Hey, doll lady!"

"Mmm? Oh, I remember you. You're the ice fairy who helped me test my Goliath Doll. Thanks to you, I got a lot of valuable data...though luckily we didn't need it since that giant youkai never appeared. Well, anyway, it was nice seeing you."

"Wait, I have a question for you!"

"Oh, you weren't just saying hello?"

"Yeah. You're smart, aren't you? I've got a math question I can't answer."

"Hmm...I suppose it wouldn't hurt. What is it?"

"How many angels can dance on the head of a pin?"

"How many...what?"

Alice was probably very much in tune with her dolls, Daiyousei thought, to be able to make an expression as blank and doll-like as that.

"How many angels can dance on the head of a pin!?" Cirno repeated, stamping her foot in the air. After all, it wasn't like she stuttered or had a weird accent that made her hard to understand.

"I don't think that's a math problem, Cirno."

"Huh? Why not?"

"Well, it sounds something more like a koan."

"That doesn't make any sense at all!" Cirno protested.

"What's a koan?" Daiyousei asked.

"Geez, Dai, even I know that. It's one of those pointy things that's round on the other end, like they put ice cream in. And some people call me an idiot," she muttered under her breath.

"Er, no," Alice said. "A koan is a kind of riddle or statement that the Buddhists use to meditate on, to help lead them to enlightenment."

"But what does it mean?"

Alice shrugged.

"I don't know. I'm not a Buddhist. If you're really curious, maybe you should go ask at Myouren Temple."

Cirno smacked her fist into her palm.

"I'll do it! No dumb ol' Buddha is going to get away with keeping secrets from me!"

"...I don't think that's quite how that works, Cirno," Daiyousei said, but the ice fairy was already flying off.

The flight to Myouren Temple didn't take too long; apparently Cirno was interested enough in finding out the answer to her question that she didn't get distracted by any curious sights they happened across, and the fairies soon found themselves flying up to the gates of the temple. The place was bustling with activity, with numerous youkai monks flitting about on one errand or another. Daiyousei wasn't quite sure that this was what a Buddhist temple was supposed to be like, but then again most of the time when this many people were together in the Hakurei Shrine there was a drinking party going on, so she supposed maybe the problem was that she really didn't understand religion.

Byakuren Hijiri was in the main temple hall, reciting something from a scroll. The air here, unlike out in the corridors, was filled with a solemn stillness, as the only two other monks present were nearly silent as they went about their chores. The voice of the nun-turned-magician echoed off the rafters in soothing fashion as she recited the sutras.

"Hey! You! Monk lady!"

The sudden screech cut into Byakuren's concentration. She fumbled the scroll, desperately tried to grab it, but only caught one spindle with her left hand; the right-hand spindle bounced off her fingertips and went rolling away across the temple floor, unrolling a good thirty feet as it went. Byakuren turned, looked up, and smiled at Cirno.

"Good morning, Miss Fairy. How may I help you?"

Daiyousei's jaw dropped. A saint! She must be a saint!

"Alice told me you'd know the answer to a cone!"

"A koan," Daiyousei hissed.

"I just said that, Dai."

"No, you...never mind."

"There are no proper answers to a true koan," Byakuren said. "They are tools by which we can focus our minds and come closer to the Buddha-nature."

"But it's a question. I mean, I mistook it for a math problem, so how can it not have an answer? That's just plain silly!"

A tiny furrow appeared between Byakuren's eyebrows.

"Why don't you tell me this koan, and perhaps I can help to guide you to the enlightenment you seek?"

"Yeah! Enlightenment means answers, right? So...okay, how many angels can dance on the head of a pin?"

Byakuren blinked.

"How many...angels?"

"Yeah! Angels!"

She tapped her chin with one fingertip.

"I don't think that actually is a koan, Cirno. It does sound like something similar, in that it appears to be an esoteric and philosophical question, so I can see why...Alice, did you say?...thought that it might be one, but in reality I think that it comes from another tradition entirely."

"Gah! You're making my head hurt!"

"She says we have to ask someone else," Daiyousei translated helpfully.

"It isn't a Buddhist saying; I can assure you of that. Perhaps it comes from one of the other religions? One of the miko might be able to help you."

"Okay. Thanks, monk lady!"

"Well," Byakuren murmured as the fairies flew away, "that was rather refreshing. Perhaps I'll have Shou ask the kappa if they can install air conditioning in the temple building?"

~X X X~

Reimu Hakurei was out in front of Hakurei Shrine when Cirno and Daiyousei arrived, sweeping dust and twigs off the veranda. It had been a particularly windy night, and dirt and debris had been scattered everywhere. Cirno's icicle wings stirred up a cloud as she buzzed in to a landing.

"Just what this shrine needs," Reimu sighed. "More fairies."

Daiyousei bowed politely. She still remembered the one time she'd had a spell card battle with Reimu.

"We're sorry to bother you, miss, but we need your help."

"Yeah!" Cirno spoke up. "Byakuren said that it wasn't a Bhuddist cone-thing, so we needed to go ask some other place!"

Reimu's gaze slid back over to Daiyousei.

"She means a koan. We're trying to figure out what it means."

"Yeah! And they're mean up at the Moriya Shrine, so we came here."

Reimu decided that she liked the idea that someone thought she was the nicer option than Sanae, even if it was a fairy (and even if it was really because of Suwako).

"All right, ask away."

"How many angels can dance on the head of a pin?"

"Huh?"

Cirno tapped her foot in the air impatiently.

"Like I said: how many angels can dance on the head of a pin?"

Reimu sighed. She then decided that wasn't quite good enough to express her feelings, so she facepalmed.

"That doesn't sound like a riddle; it sounds like a math problem."

"That's what I said!" Cirno said proudly, sticking her chest out. "But Alice said it was a koan. But Byakuren said she'd never heard of it!

Reimu nodded. "Well, neither have I." She frowned, thinking. Having offered to help, she wanted to find a solution to the problem. The last thing she wanted was to find herself stumped by Cirno's riddles, of all things. "All right, I think we ought to ask somebody who's good at math and philosophy."

"Is it a long way away?"

Reimu stood up and stretched. "Well, I think we need to wait until after I have a bath."

"A bath?" Daiyousei boggled, but Reimu was already shooting her left hand up and back behind her head in a shoulder-twisting arc, vanishing into the tear in space that had suddenly appeared and came back with her hand fisted in the front of a purple dress, hauling a tall blonde in a mobcap out onto the hisashi.

"I'm getting too predictable," Yukari mumbled into the boards.

"Seriously, how do you do that?" Reimu sighed. "Ah, well, since you're here, make yourself useful."

Yukari sat up and clucked her tongue.

"So brusque today, Reimu. And by the way, it's actually a Christian saying."

"Wait, how long have you been listening?"

"And Alice was correct to liken it to a koan, in that it is meant to be a philosophical riddle that leads one to enlightenment."

"But I wanna know the answer!" Cirno burst out.

"She has been wondering about it all day," Daiyousei pointed out. Given how long a fairy usually was able to keep a single thought in their head, that actually was fairly serious.

"Hmmm," Yukari pondered, then glanced at Reimu. "Well, since Miss Cirno is so serious about matters, it would be appropriate to find out."

"How are we supposed to...wait, are you suggesting...?"

Yukari's grin was positively evil.

"Precisely, my dear Reimu. The only way to properly answer that question...is to experiment." She cracked her knuckles.

"So which one of us fetches Tenshi, and which one of us gets the pins?"

~X X X~

A/N: I admit it, Yukari attempting to peep at Reimu is both obvious and getting old, but that segment also made my wife laugh herself silly, which by definition means it stays in! Also, those of you have been around since previous chapters will note that I finally surrendered and edited the name. It's supposed to be "The Power of Math! (feat. Yukari Yakumo)," but fanfiction-dot-net does not allow periods in its story titles (I think it may screw up their scripts for generating the URL for the story link, in all fairness...though the deletion of the second exclamation point is just them being annoying!), so "feat" without the period kept just bugging the heck out of me for its lack of grammatical precision because I am uptight about that stuff ("The Power of English!") so I gave in and changed the title to spell out the word, even if it screws up the allusion to music listings I was going for.