Cartographical. I forgive you for not loving corgis as much as I do. And you get the credit for the plot of this one. Seeing as you basically thought up most of it, and if you'll recall the circumstances if its inspiration, you'll know why I find it so damn funny that you thought up this.

This picks up mid-episode, right after the scene in which Penny tells Beckett to keep her eyes peeled for someone named Alexander.

Chapter 34: 3x2, He's Dead, She's Dead

If you don't believe in even the possibility of magic, you'll never, ever find it.

Castle's late to the precinct. He spent the morning comforting his mother over Chet's death. He knows Beckett will understand. She always does. It's one of those things about her that just makes him want to tell her –

Well.

He appreciates it.

The elevator doors open on the fourth floor, and Castle steps out. His eyes immediately find Beckett, who's back in the break room. She has an odd expression on her face. She's watching Penny Marchand, the victim's daughter, who's headed for the elevator, clearly on her way out.

Strange.

Penny sees him and smiles as she presses the elevator button. "Good morning, Mr. Castle."

"Morning, Miss Marchand. What brought you in so early?"

"Just had to deliver a message." The elevator doors open, and she steps inside. "I'm sorry to leave so quickly. Lots to do. But it's nice to see you, Mr. Castle." She steps onto the elevator, then pauses, turning back to him and holding the elevator door with one hand. "Wait. Mr. Castle?"

"Hmm?"

"When you give it, she'll give it back."

"What?"

Penny wrinkles her nose. "I'm not sure what it means."

"When I give what? To whom?"

"To her." Penny points back at the break room, where they can see Beckett stirring coffee. "I don't know what, though."

"Oh." Castle's not really sure what to think of it. Not like Penny could possibly know –

"I'm sorry I can't be clearer." She shakes her head. "It's a little jumbled. But there will be a gift. You'll give her something. Something important. It was a part of you once, and you've wanted to give it to her for a long time. At first you'll worry that she won't accept it."

"Uh. Okay?" Castle's not really sure what to make of that. He wasn't really planning on giving her a wisdom tooth.

"She'll accept it. She will." Penny furrows her brow. "Maybe not – at first. I think. But yes, I see a definite aura of acceptance. That I do know."

"Well – I'm glad to hear it – " He'll still make sure there's a gift receipt with her next birthday present, but that doesn't seem like something Penny needs to know, and he still doesn't understand how a gift can be both accepted and given back – "but I really don't know exactly what you're talking about."

"I'm sorry. It's like – there are more layers to this. There's a much bigger story ahead, but it's mostly in shadows. I can't see everything." The young woman smiles at him brightly. "But you have to solve. That's a connection. To the gift. Solve."

"Solve?" He'd ask solve what? - but really, at this point he's not sure it would matter.

"Yes." She nods vehemently. "And timing. It'll never be right. You won't think it's a good idea. But don't wait."

"Solve – something – and bad timing?"

"Exactly."

"I don't understand."

Penny frowns in concentration, shuts her eyes for a second. "All I – wait." She brightens. "One last thing. I see – sky. No. Three skies. First night. Then – sun – oh, dear – " she swallows, her face darkening for a second – "but then rain. That's it. Night, sun, and rain."

He stares at her, trying to form another coherent question beyond Are you seriously just making all of this up?, but Penny beams at him. "Have a good day, Mr. Castle." And she leaves before he can ask her what any of it meant.

Castle shrugs. Shakes it off. Whatever her mother had, Penny is obviously trying very hard to channel.

(It's not like Penny actually knows about – well. Anything.)

"My given name is Richard Alexander Rodgers. What a coincidence, huh?"

He takes one last look at Beckett (he is not admiring the soft warm glow of sunlight on her hair) before he walks away, hands bunched in his pockets.

Strange day. Strange case.

It strikes him as odd that Beckett was so fixated on his middle name. His former middle name, anyway. It's been years since he's used the initials RAR. Although he's utterly delighted that she obviously visits his website.

(And now that he thinks about it, he still doesn't know her middle name. How many uniforms will he have to bribe to get into her classified personnel file? Surely not too many. Oh, if it's Nicole he is going to just lose it. And maybe throw a party to celebrate his genius.)

He takes the stairs, not seeing anyone he walks past as he concentrates on channeling enough psychic power to divine Beckett's middle name. Elizabeth. Rose. Sarah. Sophia. Genevieve. Lucy. Lucretia? Oooooh, Natalia? He kind of desperately hopes it's Lola or Margarita or, oh, maybe it's Jezebel.

Oh, no. He wants it to be something hotter. Something like Scarlet. Would she notice if he just started calling her that? Maybe he can warm her up to it.

It's not until he's outside the precinct that he remembers solve.

One solves for x in math.

The gift was once a part of him.

His middle name used to be Alexander.

Alexander has an x in it.

He's gotten enough xoxo's beneath phone numbers on cocktail napkins to know what x can mean.

(You've wanted to give it to her for a long time.)

He almost trips on the sidewalk, stumbling like an idiot, but he ignores the curious looks he gets. He can't resist. He pulls out his phone. It's bugging me. What did she tell you about Alexander?

He hits send and waves for a cab. As he slides into the backseat, his phone buzzes. Nothing you need to know. Night, Castle.

Thanks, Beckett. He can't help but laugh, some of the tension lifting from his shoulders. So much for mystical equations that need solving. If he tried to explain any of this, she'd shoot him one of those looks that says Castle, I so completely do not have time to deal with you right now. He loves those looks. He works hard to earn them.

He walks into the loft to find Alexis doing homework. He flicks through the mail, presses a kiss to her hair. "How was school, Pumpkin?"

"Good. I spent the entire day taking notes in haiku."

"Why?"

She shrugs. "To see if I could."

"And?"

"It actually worked most of the day."

"Glad to hear it. I expect villanelles tomorrow, Princess Castle."

She laughs. "What, no sonnets?"

"Well, go with whatever moves you." He looks down at her notebook. "What are you working on here?"

"Calc." She sighs. "It's fine. It's just a lot of problems."

"Well, I have no doubt you'll be error-free as usual." He scans the page and something catches his eye. It's been way too long since he studied math. "What does that mean? Why is x pointing to infinity? Numbers can point now?"

Alexis laughs. "Dad. That's a limit. It's the limit of this function – " she points at a complicated-looking expression – "as x approaches infinity."

Ooooh. Intriguing. He likes that turn of phrase. "Approaches? What's the mathematical take on approaching infinity?" Infinity, to him, has always been the vast endlessness of space, a cloud, something you sink into, something that wraps you and holds you close and unfurls the entire universe around you forever and ever and always. In his world, the world of words, infinity is everything and everywhere. He never thought someone might have found a footpath to it.

Alexis sets her pencil down and thinks for a second. "It means how the expression behaves as we go through different values for x." She points at a diagram in the textbook, a long graceful curve that sweeps over a tidy graph. "X is a variable, so it can stand for anything. And y changes, because its value depends on the value of x."

He thinks he understands. "So you can watch the result, as x grows and develops? Even when x starts small. It grows. It changes everything."

"Yeah."

"All the way to infinity." He pats her head absently, not really noticing. "X can be everything. From one to the end."

"Dad." Alexis plucks his hand off her head and smoothes her hair. "Infinity has no end."

But it has a beginning.

He leaves her to her work and heads to his office, where he sits back in his chair, nursing a scotch and thinking. The beginning of x. If x is going to reach infinity, it needs to start somewhere. It needs to be something.

(Something he's been wanting to give her for a long time.)

A possible value for x.

But just one of many.

When he hands over her coffee the next morning, she accepts it with her usual smile and takes a long drink before she stops short. "Castle? What's this?" She looks up at him in confusion. "Is this some kind of code?"

She holds up the cup, which he has completely covered, bottom to brim, with hundreds of x's.

"Not a code." He smiles. "A solution."