WASHINGTON — In the months leading to Tuesday’s Supreme Court arguments on same-sex marriage, teams of gay rights lawyers and their allies have held countless strategy sessions, drafted scores of briefs and participated in intense moot courts.

Their relentless preparation has two goals. One is to win. The other is to win big.

“Many roads lead to marriage,” said Susan Sommer, a lawyer with Lambda Legal. “But some roads would be even better than others.”

The lawyers scoured the transcripts of arguments in earlier gay rights cases, honing answers to questions that had thrown other lawyers. They visited the Supreme Court to observe unrelated cases, taking in the rhythms of the questioning and assessing the justices’ habits of mind.

At the moot courts, they peppered the two lawyers who will argue for their side with tough, sarcastic questions modeled on the ones Justice Antonin Scalia is prone to asking.