WASHINGTON » Why did the Supreme Court agree in December to hear a major same-sex marriage case and then seem to think it had made a terrible mistake on Tuesday when it came time for arguments?

The answer lies in the gap between two numbers. It takes four votes to hear a case and five to decide one.

If nothing else, this week’s arguments provided a telling glimpse into the process through which the court selects its docket, one that is usually shrouded in exceptional secrecy. The arguments also cleared up most of the mystery of whose idea it had been to hear the case, a challenge to Proposition 8, California’s ban on same-sex marriage.

As it turns out, it would seem the conservative members of the court, making a calculation that their chances of winning would not improve with time, were behind the decision to take up the volatile subject.

The aha moment came on Tuesday.

After Justice Anthony M. Kennedy suggested that the court should dismiss the case, Justice Antonin Scalia tipped his hand.

"It’s too late for that now, isn’t it?" he said, a note of glee in his voice.

"We have crossed that river," he said.

That was a signal that it was a conservative grant.

To see why, it will help to review the bidding. When the justices gathered for their private conference on Dec. 7, they had many choices.

For starters, it was virtually certain that they would agree to hear one of several challenges to the federal Defense of Marriage Act of 1996. Two federal appeals courts had struck it down, and the court almost always reviews decisions from lower courts invalidating federal laws.

The question there, moreover, was the relatively modest one of whether the federal government must provide benefits to same-sex couples married in states that allow such unions. The case did not directly concern whether there is a right to same-sex marriage in other states.

So the justices chose one case on the 1996 law, United States v. Windsor, No. 12-307.

They then confronted a second, much more ambitious case, Hollingsworth v. Perry, No. 12-144, concerning whether the Constitution guarantees a right to same-sex marriage. Most observers thought the court would hold the case while it worked through one on the 1996 law, and some thought it might deny review, letting stand an appeals court decision that had struck down Proposition 8.

Instead, the court granted review in the case. That was a surprise and a puzzle. Who had voted to hear it?

One school of thought was that the court’s four liberals were ready to try to capture Kennedy’s decisive vote to establish a right to same-sex marriage around the nation.

That theory was demolished in the courtroom as one liberal justice after another sought to find a way to avoid providing an answer to the central question in the case. The decision to hear the case, it turned out, had come from the other side.

Scalia, almost certainly joined by Justices Clarence Thomas and Samuel A. Alito Jr., apparently made a twofold calculation: that their odds of winning would not improve as same-sex marriage grows more popular and more commonplace, and that Kennedy, who is likely to write the decision in the case concerning the 1996 law, would lock himself into rhetoric and logic that would compel him to vote for a constitutional right to same-sex marriage in a later case.

It is not that the conservatives felt certain they would win. It is that their chances would not improve in the years ahead.

That leaves the question of the fourth vote. The most likely answer is that it was that of Chief Justice John G. Roberts Jr., though he did not sound at all pleased on Tuesday to have the case before him.

There is also a chance that the fourth vote came from Kennedy himself, and his very questioning provides support for that theory.

"I just wonder," he said, sounding a little plaintive and a little angry, "if the case was properly granted."

According to "Supreme Court Practice," the leading manual on Supreme Court procedure, it is bad form for a justice who voted to deny a petition to thwart a decision on the merits.

"The reason strikes deep," Justice William O. Douglas explained in 1952. "If four can grant and the opposing five dismiss, then the four cannot get a decision of the case on the merits."

There may, of course, have been more than four votes to grant review. Tallies are never published by the court, but they do emerge years later in the justices’ papers. Historically, the court agreed to hear between a quarter to a third of its cases by just four votes.

The decisions are made on most Friday mornings during the Supreme Court term, when the nine justices meet in a private conference to vote on which cases to hear. No one else is allowed inside the room.

Under the court’s internal rules, a case is added to the docket if four justices agree. Decisions to hear cases — to grant petitions for certiorari, in court lingo — are announced in terse orders, without explanation or an indication of who voted how.

Margaret M. Cordray, a law professor at Capital University in Columbus, Ohio, who has studied the process, said it lacks most of the qualities of traditional judicial work, including deliberation, accountability and majority rule.

In examining the papers of Justices Harry A. Blackmun, William J. Brennan Jr. and Thurgood Marshall, though, Cordray said she did find "pretty strong evidence that the justices act more strategically in high-profile cases."

The court dismisses cases as "improvidently granted," or "DIGs" them, a couple of times a term. But that outcome is unlikely here, said Dennis J. Hutchinson, who teaches law at the University of Chicago and is an authority on the court.

"If they DIG it now, after all of the fanfare and all of the attention and all of the amicus briefs," he said, "it will look like they didn’t know what they were doing at the outset."

Still, said H.W. Perry Jr., a law professor at the University of Texas, "it is a case that allows a lot of exit points." The one that seemed most attractive to a majority of the justices on Tuesday was that supporters of Proposition 8 did not have standing to appeal from a decision entered against state officials.

A dismissal on standing grounds could lead to some messy follow-on litigation in California but will probably effectively allow same-sex marriages there.

So why did the court agree to hear the case?

"The justices can’t quite resist getting involved in major cases," Perry said. "This is going to come out quite unsatisfactorily to a lot of people."

Who is to blame? That will have to await the release of the justices’ papers many years from now.

"We won’t know," said Michael C. Dorf, a law professor at Cornell, "for another 50 years."

© 2013 The New York Times Company