Chief Justice John G. Roberts Jr. asked a five-word question early in oral arguments Wednesday that summed up the quandary facing the Supreme Court in a major test of affirmative action in university admissions.

He did so after Bert Rein, the attorney for a white student denied admission to the University of Texas at Austin, described how a school could determine if it has achieved diversity in its student body. The school, although prohibited from using quotas, could measure whether diversity reached “a level of critical mass,” Rein said.

“How do you do that?” Roberts asked.

Rein acknowledged the dilemma. “It’s not easy to do, and it’s not our job to do it,” Rein said. “You, the university, if you want to use this forbidden tool, this odious classification, you’ve got to find a way to do it. You’ve got to be able to explain what your concrete objective is.”

Kennedy Seen as Key Vote

The justices are grappling with how colleges can improve diversity — and measure success without relying on quotas — in a case testing Texas’ use of race in undergraduate admissions decisions. Roberts’ questions illustrated how difficult the balance is for universities trying to improve diversity on campuses, but Justice Anthony M. Kennedy is expected to be the key vote in the outcome.

Abigail Fisher, the student denied admission, is back before the court for a second time on the issue. The justices in 2013 sent her case back to the appeals court, which reconsidered it and again ruled in the university’s favor.

Eight justices heard the case Wednesday because Justice Elena Kagan has recused herself. A 4-4 vote would uphold the appeals court ruling, a result the Supreme Court likely would want to avoid.

The court’s conservatives on Wednesday appeared ready to strike down Texas’s admissions plan as unconstitutional while liberal justices stood behind the use of race to achieve campus diversity.

But Kennedy and other justices also left open the option of sending the case back to lower courts to collect more information about the plan. Kennedy lamented the absence of information that could aid the justices decision.

Roberts also pushed advocates of the admissions process to offer a way out of the difficult task.

“How does the university know when it has achieved its objective?” Roberts asked Solicitor General Donald Verrilli, who argued on behalf of the Obama administration in support of Texas’s plan. “At what point does it say, okay, the plan has worked?”

Verrilli suggested classroom studies, well-designed surveys of student attitudes, graduation rates and number of racial incidents. Roberts dismissed surveys as “sophomoric.”

The Texas university fills about 75 percent of the class by admitting the top 10 percent of each high school’s graduating class, and selects the rest with what the university calls a “holistic” approach that includes consideration of race.

Height of the Bar

Justice Sonia Sotomayor worried that the Supreme Court would use the case, Fisher v. University of Texas, to set an impossibly high legal bar for a university to prove it needs to use race to get a more diverse class. “I fear something,” Sotomayor said. If the need for proof is too high, “will any holistic review ever survive?”

Justice Samuel A. Alito Jr., the most vocal justice opposing Texas’ plan, repeatedly pointed out that Texas had not proven that the policy had made a difference, or contributed to classroom diversity.

“That does seem to me to be something that could be measured,” Alito said. “And maybe there’s evidence in the record that measures it. I don’t know.”

Kennedy used the moment to express frustration that such information isn’t included in the case.

The court first dealt with the issue in 2013, in a case now called Fisher I, by voting 7-1 to send the lawsuit back to the U.S. Court of Appeals for the 5th Circuit in New Orleans. The decision directed the court to use a tougher legal standard to evaluate the legality of the university’s admission practices. The circuit court again ruled in the university’s favor, setting up the case’s return to the high court as Fisher II, without any additional fact-finding.

“We’re just arguing the same case. It’s as if nothing had happened,” Kennedy said. “It seems to me that Justice Alito’s question indicates that this is the kind of thing we should know and we don’t know.”

Justice Stephen G. Breyer pointed to amicus briefs filed in the case that suggest universities are worried that the Supreme Court will kill affirmative action through a death by a thousand cuts, or with such a high legal bar for universities that it would effectively ban affirmative action.

“We promised in Fisher I that we wouldn’t,” Breyer said. “That opinion by seven people reflected no one’s views perfectly. But that’s what it says: Not fatal in fact.”

The case gives the court options for ruling narrowly or broadly. On one hand, a decision in favor of Fisher could threaten to bring an end to affirmative action; on the other, it could focus only on parts of Texas’s plan.

The justices will decide the case before the end of the term in June.