This was a fun question to think about. My answer talks about deepening friendships quickly, but it also applies to becoming closer with romantic partners and work colleagues.
Summary: do nice things for each other, share unique experiences, have meaningful conversations, keep your guard down, support each other through difficult times, act as if you were already close, and listen carefully.
1) Do favors for each other. Studies show that when you do something nice for someone, especially on a whim, then you will like them more. That's right, the person doing the favor will feel closer to the person receiving the favor. It turns out that when we go out of our way to do something for someone else, our minds thinks, "I did this nice thing that I didn't have to do, so this person must be worth that effort." So, when you get a chance, go out of your way to visit a friend at the hospital, help them move to a new house, stay up late helping them practice for a presentation, and so on. Conversely, let people do nice things for you, even if you think you can manage by yourself.
As an anecdote, I needed four friends to help me out with a personal matter a few years ago. One of them bowed out at the last minute due to a family emergency, so I asked one of my coworkers, John, to fill in. He did, at considerable inconvenience to himself, and I was very grateful. Our careers diverged shortly thereafter, but we've kept in touch regularly. Even though we live on opposite coasts of the US, we feel pretty close because of that single favor. If John hadn't helped, we probably would have lost touch because, initially, we were not very close.
2) Share unique experiences. If you go watch a movie or have dinner with someone, it's just another evening. Three years from now, you probably won't remember the day any more than you'll remember what you did the day before or the day after. However, if you go to an amazing concert or eat at the French Laundry or hike up Half Dome for the first time, you'll remember the event vividly and, by association, you'll remember the people you shared it with. When someone is a core part of many of your most precious memories, it's hard not to feel closer to them.
3) Have meaningful conversations. The "10,000 hours of practice" meme suggests that enough deliberate practice can make you a world-class performer in almost any field. The "deliberate" part is key. If you repeat the same tennis shots for three hours a day over the course of a decade, you'll definitely be a very good tennis player. To be world-class, however, you need to practice thoughtfully. You have to challenge yourself with different shots, try small adjustments to your stroke, and so on. The goal is thoughtful repetition, not mindless repetition.
Conversations are the same. You can talk about a TV show that you like or the Giants' World Series win or how bad traffic was this morning, but it probably won't lead to a great bonding experience. Instead, talk about what you want out of life, how you felt when you didn't get accepted to the college you wanted to go to, what kind of parent you want to be, and so on. Obviously, you need to strike a balance of meaningful conversations and regular conversations, but the meaningful ones will really accelerate your friendship.
4) Open up. This is a combination of #1 and #3. When you're open with people, you'll feel closer to them. Furthermore, most people will reciprocate and talk more openly when you're being honest and vulnerable. This will, in turn, make them feel closer to you. It's a great positive feedback loop.
I recently went through a significant life event, and I was initially hesitant and embarrassed to talk to others about it. I'd gloss over it in casual conversations, then quickly move on to the next topic. Then I did something different: I tried being more open. To my surprise, a lot of great discussions came out of that. I learned that many of my friends experienced similar issues, we talked about how we dealt with things -- or how we wished we had dealt with them -- and we became much closer as a result. The things we believe are too shameful to talk about are rarely anything to be ashamed of.
5) Suffer together. Do something that each of you believes will be hard. This might mean volunteering for a presentation when both of you are afraid of public speaking, training for a marathon, or taking a college class together. When two people suffer side-by-side, the suffering deepens their connection.
6) Fake it 'til you make it. Think of all things you might do with your close friends: you drink from the same glass, try each other's food, aren't afraid to be yourselves, act silly, call and text frequently, take turns paying for meals without keeping track of a few dollars here and there, and so on. These things come from being comfortable with each other, but they can also be used to become more comfortable with each other. You become closer by acting as if you were already close.
One study showed that pairs of strangers who were asked to stare into each other's eyes for several minutes reported higher feelings of affection and attraction than a control group. Pretend intimacy results in actual intimacy.
7) Be an attentive listener. What you do and what you don't do can be equally important. If you optimize the hell out of the activities you do together but forget important things about the other person, you will end up sabotaging your own work. Nothing communicates "I don't care about you" more than you not remembering something that's very meaningful to your new friend. I learned this lesson the hard way. If you have to, jot down a few notes after each important conversation.
These tips are not for everyone -- for example, if you don't like sharing food, then you don't like sharing food -- but they have all been very helpful in my own life.
A final caveat is to respect speed limits. Sharing is good, openness is good, and unique experiences are good, but that doesn't mean that your first meeting should involve telling someone about your issues with your parents while you awkwardly stare into their eyes during a treacherous hike. Pace yourself.
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