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 that 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.