Their relationship can be read through the games they play.
At first, he plays solitaire. Solitary. It's in the name, a game for one person, one mind. He doesn't have the patience for interacting with the others and he doesn't trust Zaimokuza to sit still for long enough or Totsuka to NOT distracted him so he can put his full focus into a two-player game. So when there's downtime, he finds himself a corner near the tennis field, during what should be the club time, and lays out a deck. While his solitaire skill is not that great, he still can say he can beat 90%, no, 80% of the school population, including the teachers.
He was kind of tempting fate, playing it out in the open. If he had truly wanted to play alone, he would have kept the cards in his pocket until in the privacy of his room. But he didn't. On some subconscious level, Hikigaya Hachiman was looking for an opponent; not an enemy(riajuu) to cut down or some kind of partner(unfortunately) like Zaimokuza, someone in the middle, with the fight of an enemy but with the warmth of a rival.
Not that he realized that. On a superficial level, he just wanted something to pass the club time away and solitaire did just that.
Her shadow passes over him as she watches over his shoulder. An arm creeps around and a hand that definitely wasn't his (unless he'd taken to painting his nails pink-red, which he highly doubted) picks up the seven of clubs and deposits it with the other club cards gathering over their ace. Hachiman narrows his fish, no, dead fish eyes; he hadn't noticed that which probably annoyed him more than the fact SHE was getting involved in his game.
"This is a game for one, Miura." He drawls, batting her hand away the second time it approaches.
Yumiko's shadow disappears and he assumes she had left, but then she appears in front of him, sitting cross-legged across from him. He realized he can see a bit of her pink panties when she sit like that. Without a word, she scoops up his game of solitaire and begins shuffling, despite his flabbergasted look. "So let's start a new one."
She drops his deal of the cards in front of him. He glares at her but picks them up anyway.
Why is she here? Doesn't she have a club to attend? Ah wait, she is part of the "going home" club. Just like him once, until the nicotine-loving teacher of his forced him into the Service Club.
Their tolerance for each other begins with poker. Neither trust the other, so it's the perfect game. Behind carefully stoic faces they both watch the other, neither wanting to be the first to crack and actually begin something resembling friendship. No, they're both too stubborn for that, so they sit with blank masks on and stare each other down.
She catches him slipping an ace from the sleeve of his jacket and she throws her cards at him.
He grins and it's the first bit of expression she's ever witnessed from him that's not cynical, sarcasm or condescension.
He briefly suggests strip poker. Her glare lets him know their relationship is thawing, but it isn't there yet.
But she doesn't mind him seeing her panties whenever they played on the floor. Hmm, black lace today?
Their friendship begins with gin. It takes less concentration that poker doesn't require as much bluffing and wearing of masks, so they start to loosen up. They no longer sit in stony silence other than to trade barbs and remarks over the fan of their cards; now they can chat casually as they rearrange until one of them gets the patterns they need to win.
They begin trading stories for wins; she triumphs and he tells her about Komachi, about the Service Club's exploit, and before long after, she could tell him exactly what he had for his first meal after his first dentist or how he got the scar winding around his forearm.
His track record is as good as hers, even if he rarely put his all, so she paints him stories of afternoons in the backyard with how she met Yui, tales of an embarrassing story about Ebina, the popular game within the circle of high school and college students, tell him how it felt when she first woke up after her first drunk experience and wasn't completely there.
She's just put down three nines and a climbing row of diamonds when he realizes he's running out of things to tell her. When she looks at him expectantly, he puts down his own cards (he needed one seven and he would have won) and gives her a look that she would almost call nervous. A look that she rarely sees.
"Let's not talk about the past. I've been thinking about the future."
Their romantic relationship comes back to solitaire.
They no longer need cards games to have an excuse to spend time with each other. Fingers brushing as they exchange a card isn't the only way they can get physical contact anymore. Conversations begin without the banter of the game comes first. Little here, little chat there.
He no longer plays his solitaire out in the open either - now that he is no longer looking for an opponent, they do it in the privacy of his/her bedroom, the school toilet, the sports equipment room, or at the empty classroom.
He only needs to kill the time before she is awake for him to commit his attention to and be focused on in return; while he waits, he plays his cards over her bare back as she dozes in his bed.
"That tickles," Yumiko mumbles childishly as he deals across her skin.
He smirks and brushes her hair over her shoulder so it is out of his way.
Halfway through, she rolls over to face him, spilling his game across the sheets. He'd be more annoyed, but the fact she moved so she could sit up and kiss him dissipates it a little. She pulls him back down with her and all annoyance is forgotten.
Later she pulls a crumpled ace of hearts out of her hair and hands it to him triumphantly. He flicks it away and goes in for round two.
(Everything they do until now has nothing to do with Hayama and Yukino's arranged engagement. There is no way they did this to soothe their broken heart. that just their imagination.)