James and Lily's relationship was like a wooden brainteaser.
It started out much the same; when you first saw it, it looked like the simplest shapes had simply been glued together. It isn't until you've flipped it around in your hands and pushed it and pulled it as much as it would allow that you realize it's much, much more complicated than that.
(Lily thought James was just a big, bullying jerk, no questions asked).
You have the two key pieces, constantly engaged, never budging an inch, with smaller, seemingly less important pieces that can move the tiniest amounts that don't seem no matter in the slightest. It all seems impossible, until you notice the tiny square that you hadn't noticed between the two can be slipped out via a hole in one and a jut in the other.
(Snape was the biggest factor keeping them apart. But a bumpy day, courtesy of James, made him disappear with the one word, from the one person, Lily didn't have a guard up against).
The tiniest, most seemingly inconsequential nudges can make the biggest difference.
(Lily had never seen James apologize for anything before. But to her?... such a thing was completely unheard of).
Sometimes, before the pieces can be solved, you have to pull them in opposite directions and just let the knots around them work out.
(Winter Break meant Sirius finally severing connection with his family and staying with James. It meant Petunia simplifying matters considerably by getting married and moving out. It also meant time apart for Lily and James to cool down after weeks of arguing).
Whoever makes the puzzles certainly loves to confuse you by painting different pieces the same color. You're so tempted to think they're just one piece, or at least similar pieces, no matter how illogical that may be.
(Lily realized over a Valentine's Day trip to Hogsmeade that just because Connor McLaggen was a Gryffindor Quidditch player with black hair who loved to have fun didn't make him James, though she didn't use quite that wording.)
These puzzles are all about experimenting. You have to try one move, and be constantly aware of how that affects all of the other pieces.
(Lily dating McLaggen left James moody and volatile. After breaking up, he seemed considerably more cheerful. Of course, if Lily actually noticed, that would have been much more significant).
No matter how much it seems like a move is going to work, sometimes it just isn't.
(Lily was thoroughly shocked to find that obsessing over her studies and reading constantly did not actually solve her "problem". James reacted similarly when a flood of pranks with Sirius had the same effect, or lack thereof).
Nothing can give you more hope than that one big move.
(A single kiss had the exact opposite effect Lily expected. She thought everything would get complicated; instead, she found herself giddy and in the unlikely pursuit of another).
Once you've caught the hang of it, the puzzle becomes a flurry of moves and solutions until—
(After that kiss, James and Lily knew they needed to talk. And so they did. Sure, not every couple gets together with a shouting match but the breakfast table, but hey, nothing they did was at all conventional).
—the puzzle was solved.
Aah, that's been stuck in my brain for ages now. Nothing makes the world seem a bit brighter like a nice Lily/James, I always say.