Hello there! I keep getting more and more ideas for the missing months between Sherlock 'outing' Mary and Christmas, so have another one. In this one, Archie keeps Sherlock company and fun times are had by both. I'm not sure how old Archie was meant to be, but I made him old enough for Algebra, because I can.

This one-shot could be read as part of the A Chair In Its Proper Place series, or not. Your choice. =)

Algebra for Detectives

For the sixth time today, Sherlock finished playing Wagner's Träume on his mind palace violin. He would have liked to play his own instrument, but the hospital staff had confiscated it. As if some classical music would disturb the neighboring patients!

With a frustrated sigh, he picked up his mobile to check for messages. There had been nothing since 12:08 PM, when John had texted during his lunch break at the new clinic. As nice as it had been to have John with him for two weeks (like old times, almost), Sherlock was well aware that John needed something to do, even if it was a dull GP position in a boring, middle-class clinic.

As he pondered the extramarital affairs of the nurse on duty, his mobile buzzed, signaling a new FaceTime call...from Archie Campbell.

He answered at once, and saw the boy seated at his desk at home. His bedroom was an explosion of clothes and books and miscellaneous stuff, but there was some order to it. In a way, it reminded Sherlock of his own flat.

"Hi, Sherlock," the boy said brightly. "Janine told Mum you were shot. I'd come down to visit, but Mum wouldn't let me go on my own."

"I was shot," the detective admitted. "Right here," he said, pointing the phone's camera so Archie could see.

"Did it hurt?" Archie asked curiously.

"Yeah, a bit," Sherlock said honestly. "But they fill you up with drugs here, so you can't feel it. Makes you sleepy, though. It's loathsome."

Archie bit his lip. "Does it bother you that I called? Should I hang up then, let you sleep?"

"No!" cried Sherlock. "I don't think I've ever been this bored in my life, Archie, and I hate it. I'm glad you called."

"Oh, alright," the former ringbearer said, smiling. "I wanted to ask how you were, and—" he paused.

"Yes?"

"Well, I don't s'pose you use a lot of algebra as a detective, do you?" Archie asked shyly. "I have some maths homework to do, but Mum can't help me with it."

Sherlock felt a strange warmth in his chest. He'd only seen Archie three times, and already the boy trusted him with his homework. "You'd be surprised," he said in a conspiratorial tone. "When teachers tell you that you'll use algebra all your life, they're mostly lying. However, it can come in handy for crime-solving sometimes, unlike history or pop culture."

"So you'll help me?" the boy asked hopefully.

"Sure," said Sherlock. "What do you need to do?"

Archie rummaged around his desk until he found what he was looking for. It was a Year 6 maths workbook, well battered after months in a backpack. The curly-haired boy turned the pages quickly, reaching the correct chapter in seconds.

"I'm to write a story problem for each of these," Archie griped, "and then solve for x. I hate story problems!"

"Oh, there are ways to make them fun," Sherlock told him, raising an eyebrow. "Read me the first one, and we'll have a go."

Archie shuffled through his papers until he found his maths workbook. "4x + 5 = 105."

Sherlock thought about it for a minute. "Right. Let's say you have a serial killer, and his name is James. That bit is true; I've met three serial killers named James."

The boy stared curiously through his webcam. "Did you beat them?"

"I did," Sherlock said smugly. "The last one was hard, and I had to fake my own death to beat him, but I managed it. Now, back to the case." He took a breath and steepled his hands under his chin, leaving the phone propped at an angle against the rail of his bed.

"James had a brother named Joseph, who was five years older. They didn't like each other very much, so James killed him eventually. Their father Alexander saw James doing it, so James killed him too, and threw the bodies into the Thames. If James' dad were alive, he'd be twice his age."

"Cool," said Archie, scribbling away.

"Now, this is important," Sherlock continued, "because James didn't stop there. You, the intrepid detective, have discovered that James always kills two people at once; one is five years older, and one is twice his age, always. Serial killers like patterns," he said knowingly, "and that's doubly true if there's a messy family history. Sentiment," he scoffed.

"If Joseph and Alexander were alive today, the sum of their ages combined with James' would make 105 years. How old is James? Remember that his age determines the victims he chooses, so this is a crucial bit of information for a detective."

"That's wicked," Archie said, fascinated. "Does it really work with this problem, though?"

"Write it out," Sherlock suggested. "You'll see."

Archie looked through his notes and turned them into an equation. "So...James' age is x, Joseph's age is x plus five, and Alexander's age is 2x. That means 4x plus five equals a hundred and five, and x is twenty-five. James is twenty-five, and his next victims will be thirty and fifty!"

Sherlock beamed at him. "Well done, Archie."

"This is so cool," the boy told the detective. "Can we do another one?"

"Of course," Sherlock said agreeably. "I'm not allowed to have real cases at the moment, so we might as well create the perfect murder for your maths homework."

Half an hour later, Sherlock and Archie's story problems were getting bloodier and more elaborate. Both were having fun, and Archie had long ceased to struggle with the concept.

"3x + 3 = 9," Archie read, waiting for more.

Sherlock thought about it, then spun his tale. "A Scotland Yard sergeant is chasing two kidnappers and their hostages. He's armed with a nine millimeter handgun with a fifteen-round magazine. He used six bullets earlier in the chase, but our police friend has rather awful aim, and they all missed."

The boy giggled at this, and Sherlock's mouth twitched.

"He fires again. This time, three of his shots miss, and he hits the first criminal twice as often as the second. When the two crooks are arrested, the copper's magazine is empty. How many times did our valiant policeman hit each kidnapper?"

"Well," Archie mused, "The gun only had nine bullets left, and three were misses. Then for the kidnappers, the second guy got shot x times, and his partner 2x. So...x plus 2x makes 3x, plus three misses, equals nine. X is two, so one man was shot twice and the other four times."

"Poor fellow," Sherlock said, wincing in sympathy. "Then again, he's a kidnapper, so who cares. Next?"

"That was the last one," Archie said, grinning. "I've never had so much fun in maths class. You should teach it."

Sherlock chuckled at this, stopping abruptly when the movement pulled at his wound. "I don't think anyone would hire me to work with children, you know. But grown-ups are stupid that way."

"Yeah," the boy agreed. "You're okay for a grown-up, though. And really good at maths, which is great."

The detective winked. "Don't tell anyone. The press thinks I'm an awful monster thanks to John's blog, and that means they don't bother me between high-profile cases. I like it that way."

"I'm not scared of you," Archie told him. "But I told my mates about the headless nun pictures, and they didn't believe me."

"It happens," the man sighed.

"Mrs. A might turn green when she sees this," Archie snickered, waving his workbook, "but I think she reads John's blog, so maybe not."

"Just tell her I'm training you to be the next consulting detective," Sherlock offered. "Maths are so much easier to stomach when there's a fun practical application, and she can't get too upset if the calculations are perfect, which they are."

"Would you really?" the curly-haired boy asked, eyes wide. "Train me, I mean."

Sherlock shrugged, as much as he could without jarring his wound. "You helped me solve the Mayfly Man case, didn't you? You're on your way already."

The man caught a brief glimpse of his young pupil's happy face, and then his phone beeped alarmingly and showed a '10% Battery Remaining' message.

"Well, Archie, it appears my phone has had enough algebra for today," Sherlock said slowly. "Thanks for calling. It's been fun."

John Watson would have been surprised at the sincerity in that statement. Then again, John had been out of the picture for years. He was still getting to know the new, post-Reichenbach, post-gunshot Sherlock Holmes.

"Yeah, thanks so much for the help," Archie said enthusiastically.

"Call me again if you need me," Sherlock encouraged him. "I'll even sink so low as to help with English or French homework if I must. I'll do anything to alleviate the tedium at this point, though murders are always preferable."

The boy laughed. "If I need help hiding a body, I'll call you. Bye, Sherlock!"

Sherlock ended the call, smiling. John would be coming soon, and perhaps he'd bring Mrs. Hudson along. He was sure that one of them would bring an adequate distraction.