#i know the upper bound is ½ with a chessboard pattern

[mowu-moment]

falling alseep to math videos and having weird dreams. Suppose the patrons of a large movie theater enter the theater one at a time, and choose a seat at random among seats that are not orthogonally adjacent to any other moviegoer (up, down, left, or right). Seats are placed on a square grid of indefinite size. What is the average proportion of seats filled before a moviegoer must sit next to another?

[annies-hoodie]

Prompt: Annie learns to rely to others after spraining her ankle and has to be carried by Mikasa through the forest by 3dmg. She's terrified of being in a situation where someone has her life in their hands and could just drop her to her death but Mikasa reassures her that she's holding her and she'll be safe.

jesus. this took forever and i truly am sorry for that. it’s a long one. hope you enjoy!

The rain hit Annie’s shoulders like pelts of pebbles, drenching her hoodie and matting her hair as she moved through the forest swiftly. She could begin to feel her grip on her maneuvering gear slacken, the water loosening her hold as beads of water slid down her palms. Still she fought to cling on tightly, refusing to let the weather get in her way of beating Mikasa to the finish line.

Today was all about going fast, or, as Shadis had called it, the imperative skill of speed . The trainees were racing in groups of four to see who could record the fastest time through the designated area of the forest, and Annie had been matched up against Connie, Jean, and Mikasa. Connie and Jean posed no problem at all. Annie knew she could out-maneuver them even at 50% effort, but Mikasa was another story. She always was. And while Annie was confident about disarming the other girl in hand to hand combat, in the air she was forced to begrudgingly accept that Mikasa had an edge over her. Her height and body weight gave her better momentum and speed when navigating, and Annie found herself having to use every ounce of her upper body strength to keep up with her. But what Annie lacked in physicality she made up with acute spacial awareness. Constantly mapping out the shortest route possible, Annie became one with her surroundings and was able to traverse even the most difficult of areas, innately knowing just how close to cut a corner to make the most of a direction change, or which pattern of trees to hook onto in order to make the fastest retreat. She often imagined she was the queen of a chessboard, zigzagging and conquering the inanimate objects in the way of her goal. Today was no exception, as she cut in front of Mikasa, narrowly avoiding a collision as she fired her grappling hooks at tree diagonal from her, and pushed off the trunk, using the momentum of her spring to push her forward and allow her to propel herself a good length away from the other girl.

It wouldn’t be long now until she’d reach the clearing and get to internally bask in her victory.

Extremely aware that even the smallest of mistakes would cost her the win, she chose not to focus on the too close for comfort sounds of Mikasa’s gear emitting tufts of gas as she increased her speed to catch up. Raindrops streamed down Annie’s forehead as the showers above them continued to pour, and her bangs became plastered to her face, partially blocking her view as she scanned for her next course of action. Every move from hereon out would need to be perfect, and executed with flawless technique or she’d find herself second to the only person that came close to rivaling her in the entire camp.

And that she just couldn’t accept.

Firing her grappling hooks to a pair of trees forward, she envisioned herself catapulting through the opening to push off a titan dummy and execute a spin, using the motion to increase her velocity as she pulled herself forward. But as she bolted towards the dummy, she realized just a fraction too late that friction would fail her, that the soles of her worn boots would most certainly slide right off the smooth surface. The rain continued to fall heavily as Annie careened off the side of the dummy, trying in vain to direct her gear elsewhere as she helplessly spun towards a nearby tree, her equipment smashing into it and forcing her to barrel towards the ground. She landed hard on her ankle, twisting it sharply as she yelped in pain. Her face splashed into a puddle of muddy water and she sputtered as the cold liquid seeped beneath her collar, struggling to sit upright after the fall had knocked the breath out of her.

Her gear lay in a tangled, useless mess a few feet from her, and tears began to well in her eyes despite herself as she saw Mikasa fly past her, and Jean and Connie soon afterward. How could she possibly complete her mission if she couldn’t even best Mikasa in a simple training exercise? And more importantly, how was she going to get back to the grounds with her ankle in this condition?

To test it, she tried to stand up and immediately sank back to the ground, her ankle throbbing with searing pain. She was going to have to examine the injury. Annie scooted closer to the tree she’d hit, hoping to gain a reprieve from the rainfall under its leaves. She gingerly removed her slippery boot, and peeled away her sock, soaked with rainwater, to reveal a large, purple area of swelling on her ankle. She closed her eyes in frustration, thankful for the gloomy weather for the first time that day as the precipitation masked the single tear that managed to escape her eyes.  There was no way she could walk on this thing. She’d have to wait until one of the nicer trainees like Sasha, or Krista, or even Reiner and Bertholt noticed she was missing, and came to find her to bring her back to camp. But relying on one of them would show weakness…and this loss to Mikasa was enough defeat for the day for Annie. She refused to be further humiliated in front of the trainees, and deepen the disappointment in herself. Maybe she’d walk back to the grounds on her own, ankle be damned. She could fight through the pain. She’d been through much worse, anyway. As long as she found a way to walk that would take most of the pressure off the ankle, she estimated she’d be able to make her way back in about thirty minutes or so, taking her hindered movement into consideration.

She began to survey the surrounding area for a pair of branches that could possibly act as a makeshift pair of crutches when she was surprised to see Mikasa hurtling towards her, her expression unreadable as she disengaged her gear and landed firmly on the ground a few feet from Annie. Mikasa walked briskly towards her, stopping just a few paces from where Annie sat.

“You’re hurt.”

Annie looked away from Mikasa, hoping to cover the sight of the swelling with her other boot. Why had Mikasa come back? She’d flew by not even a minute ago, but if she had reversed, that meant Jean and Connie would be taking the lead right about now.

Not answering her question, Annie replied while still avoiding eye contact. “What are you doing? You’re going to lose if you stay here any longer.”

Mikasa was silent a moment before continuing, taking one step closer to Annie. Annie didn’t flinch as Mikasa’s boot collided gently with a puddle, sending sputtering droplets of water into Annie’s side.

The pitter-pat of raindrops continued to sound around them like tiny drums, beating to their own tune. Annie watched as they sunk into the saturated earth, disappearing just as soon as they’d came.

“I don’t care about winning or losing a training exercise. I’ve already proven myself.” Mikasa said, and Annie looked up to find a slight inkling of concern across her features as she surveyed Annie’s ankle, poorly hidden behind her half-baked attempt at hiding it. “Your injury looks serious. We need to get you to the medic.”

Annie’s clothing felt heavy and confining on her, clinging uncomfortably to her body as it became weighed down with rain. An imprint in the mud began to form where she continued to sit. “I can make it back myself.”

Mikasa’s eyes slowly took in Annie’s rain soaked form, noticing the slight chills that reverberated through her body as the chilly, damp weather began to permeate her bones. She regarded the indignation in her ice blue eyes, and nodded. “I know you can. But I can get you there faster.”

She looked at her own maneuvering gear and back at Annie, gauging her reaction. The blonde’s eyes traveled from Mikasa to her gear, her expression hardening. “No.”

Usually Mikasa would not take no for an answer if it had been anyone else. But Annie was different than most people. Mikasa knew she’d have to reason out her wordless proposition to carry Annie back to the grounds for her to agree.

“Why not?”

Annie’s face was hard to read; Mikasa couldn’t tell if she was annoyed by the offer or genuinely angry. “Because i can find my own way. I don’t need your help.”

Mikasa’s temper flared before she realized she’d act the same if their positions were reversed. Her offer must seem condescending. She needed Annie to know she wasn’t trying to question her abilities.

“Annie.”

She waited to continue until her eyes locked again with Annie’s.

Annie scoffed. “Don’t patronize me.”

Mikasa shook her head. “That’s not my intention. I want you to trust me.”

Annie’s reply was quick as her eyes became cold, expression bitter.

“I don’t trust anyone.”

Mikasa could understand this. After all, she herself followed the same mantra. Perhaps she’d misspoke.

“Fine. I won’t tell you to trust me. But let me try and show you.”

Annie was unmoved. Trust the girl who was so quick to take her down on the training grounds? The girl who’d pummel her face in without a second thought if she had the opening? Yeah, right. “‘I’d rather die.”

Kneeling down beside Annie, Mikasa looked at her earnestly. “No. We need you.”

Annie almost laughed, a bitter edge to her voice as she spoke. “We? Does that include you?”

Mikasa was still. Of course it included her. Annie was the only person who challenged her, the only person that kept her on her toes. The only one who didn’t ask why she was so quiet all the time, or didn’t look at her with fear. Though Annie was her opposite, she sensed a sort of underlying connection with the other girl, perhaps bound by their mutual appreciation of silence and hard work  Or maybe there was more to it. At least, she suspected there was. But for some reason, she found herself unable to say those words. “Eren looks up to you.”

Annie fought the urge to roll her eyes at the expected answer. “Right.”

Mikasa could feel herself losing Annie. It was as if her words were strikes, and if she said one more wrong thing, she’d be out. Discussion over. And Annie would either stay, drenched in rain in the cold forest until someone else came to her her, or risk further injury by attempting to get back to camp herself. Both were unacceptable outcomes to Mikasa. Regardless of her ambivalent feelings towards Annie, she did not want to see her hurt. Or cold. Or alone.  “Annie. I won’t ask for anything in return for helping you. And I’m not doing this to prove a point.”

When Annie didn’t respond, Mikasa continued.

“When I say I want to help you, it’s because I want to help you and nothing else.”

Annie was no longer looking at her, her eyes fixated on her ankle. She hugged herself tightly, and shivered, eyes wincing just slightly as a crack of thunder boomed overhead. As if on cue, the intensity of the rainfall increased and Annie bowed her head, letting the raindrops drip down her chin. Mikasa thought she looked incredibly small in that moment, reminiscent of a child lost in the woods. She felt the urge to wrap her arms around Annie, to keep her warm and shield her from the rain, but the feeling was gone quickly, like the flame of a candle snuffed out abruptly. She had no room left in her heart to care for Annie in such a way. Maybe, if they lived in a different world, things would be different. But Mikasa would be a fool to entertain such delusions and ideas. She needed to focus on the present. And her present and foreseeable future included protecting Eren and Armin. That was it. To let others in would be a grave mistake, a vulnerability she could not afford to have. Not in this cruel world.

As Mikasa watched Annie, she realized that she had lost. Annie was not going to let her carry her back. She wanted to do it alone, and Mikasa would let her. As much as she hated the thought of Annie making the trek back solo, hurting herself even more and falling victim to a nasty cold, or even pneumonia, she couldn’t force Annie to do anything. At the end of the day, she was her own person. Mikasa had said her piece and made her offer. Annie had refused. And that was all there was to it, whether she liked it or not.

With one last look at Annie, Mikasa turned around and began walking slowly back to camp.

“Mikasa.”

Through the rainfall, showering down with such force that Mikasa had thought she imagined it, Mikasa turned at the sound of her name. Annie was looking at her, eyes a deep, open blue instead of their usual icy hardness.

“I’ll go with you.”

It was all Mikasa could do to not breathe a sigh of relief. She walked over to Annie and crouched down, wasting no time in wrapping her arms around the smaller girl to hoist her up. She hissed at the contact, Annie’s freezing form clinging to hers.

“Annie…”

The other girl was so cold. Her hoodie seemed to have absorbed an insane amount of rain, judging by splatters of water that had been wrung from the cotton when Mikasa had stood her upright. It also didn’t help that the puddle of mud she had been sitting in had soaked right through her pants, caushing her to shiver uncontrollably. The purple, bruised area on her ankle contrasted sharply against her pale white skin, and Mikasa wished that she had something to cover it with. But her clothing was in no better shape. She’d just have to try her best to get Annie back as fast as she could, so she could get into some warm, dry clothes.

“I’m going to hold you in one arm and navigate with the other. I’ll get you back as soon as possible,” Mikasa said, hooking her arm around Annie’s waist.

Annie stood rigid despite her shaking, bringing one hand up to hold Mikasa’s arm as if on instinct. “What if I fall?”

Mikasa tightened her hold on her. “You won’t. I promise.”

Annie knew her nervousness was uncharacteristic, but she had never been in this position in her life. She would be completely dependent on Mikasa the entire duration of the trip back, her life essentially in her hands. Mikasa could easily drop her if she wanted, laughing above as Annie snapped her neck on a stump. She could leave her in another part of the forest and continue on, abandoning Annie to her fate. Or she could fall out of her grip on accident, just like Annie had lost the grip on her footing. So much more could go wrong than right, and Annie’s heart was pounding with anxiety.

She looked up at Mikasa and saw her strong, set jaw, her grey eyes earnest as she awaited Annie’s signal. Seeing her strength made Annie feel slightly better, so without dwelling further, she nodded, ready to go.

As soon as her feet left the ground though, Annie started to panic. She felt as if she couldn’t breath, the trees a blur around her as they flew forward. Her feet dangled helplessly beneath her, and she felt the urge to cry as Mikasa swung them ahead. She couldn’t say she was scared. She’d never forgive herself. But she felt so powerless, dead weight on Mikasa’s arm. She was afraid of not having control, of having to rely on someone. Especially Mikasa, who Annie was constantly locked into some kind of battle with, whether it be physical or mental. Letting her take the lead was incredibly risky, and Annie swallowed hard as she broke into a cold sweat, the small but constant fear of falling nagging at the back of her mind.

“You’re okay. I’ve got you.”

Mikasa spoke with her eyes trained ahead, and Annie’s breathing slowed a bit, feeling her pulse gradually return to normal at the sound of her voice.

“We’re almost back. Hold on a little longer. Just trust me.”

And to her surprise, she did. As they went on, Annie began to trust Mikasa. Whenever Annie began to feel her anxiety return, Mikasa would instinctively hold her closer, or say something to quell her fears.

“Don’t be scared.”

“You’re safe.”

“I’m here.”

She would close her eyes as the black haired girl spoke, letting the calm certainty in her voice envelope her and lessen the tension in her limbs and  mind. Mikasa propelled them forward seamlessly, taking care to assure her movements were fluid and gentle so Annie would feel at ease.

Before long they reached the clearing, and Mikasa floated down to the ground with Annie in tow, the softest landing Annie had ever seen anyone execute with the gear.

When their feet touched the ground, Mikasa silently offered her elbow to Annie to help her the rest of the way to camp. Annie looked up at her, the rain subsiding as Mikasa brushed her own bangs out of her eyes. A part of her was still in disbelief that out of everyone, it had been Mikasa that came back for her.  Mikasa, who had the power to knock the breath out of her lungs, to kick her onto her back without breaking a sweat. The one who looked at Annie with cutting indifference, like she was insignificant and not worth her time. Yet in Annie’s moment of need, Mikasa hadn’t taken advantage of her, or tried to play her. In her weakened state, she offered help instead of sabotage. Trust instead of betrayal. And though Annie had taken a huge risk in trusting her, she knew now, as she hooked her arm through MIkasa’s, that it had been the right decision.

When Mikasa stole a glance at Annie on their way back, she noticed the new softness in her eyes as she kept in stride with her. Daily she reminded herself she had no more room in her heart to care for other people besides Eren and Armin, just as she had earlier. But now, looking at Annie, she thought maybe she had room for one more. And maybe, admiring the glow the setting sun cast on Annie’s features, she thought the world could be beautiful too.