WASHINGTON — The Supreme Court yesterday wrestled with voluntary integration plans in public schools, asking whether programs in Louisville, Ky., and Seattle are acceptable moves toward student diversity or other names for illegal racial quotas.

Justice Kennedy, who could hold the decisive vote, joined his conservative colleagues in expressing deep skepticism about the programs.

The Seattle district seems to be telling its high school students who are subject to the plan that “everybody can get a meal,” but that only certain people can get “dessert,” Justice Kennedy said. He was referring to the fact that some students did not get assigned to the schools they preferred based on their race. About Louisville’s system-wide assignment plan, Justice Kennedy said, “It’s a troubling case.”

The court’s four liberal justices indicated they see no constitutional problem with school districts that factor in a student’s race in an effort to have individual school populations approximate the racial makeup of the entire system. Federal appeals courts have upheld both programs.

In Louisville, the school system spent 25 years under a federal court order to desegregate its system. The school board decided to keep much of the court-ordered plan in place to prevent schools from re-segregating.

“What’s constitutionally required one day is constitutionally prohibited the next day? That’s very odd,” Justice Ginsburg said.

Francis Mellen Jr., representing the Louisville schools, called the plan a success story that enjoys broad community support, including among parents of white and black students.

Justice Kennedy wanted to know that even assuming the school board’s good faith, can it assign students “based on the color of an individual child’s skin?”

The justices and lawyers arguing the cases referred repeatedly to a 2003 Supreme Court ruling that permitted the limited consideration of race to attain a diverse student body on the college level.

Justice Roberts expressed concern about assigning schools “based on skin color” and not “any other factor.”

Attorney Michael Madden, representing the Seattle school district, said race is but one factor, that it is relied on only in some instances and then only at the end of a lengthy process.

Mr. Madden drew a distinction between the Seattle school program and the subject of the court’s 2003 decision, which narrowly approved the University of Michigan law school affirmative action admissions program.

“This is not like being denied admission to a state’s flagship university,” Mr. Madden told Chief Justice Roberts. The Seattle students are “not being denied admission, they are being redistributed.”

Amid the oral arguments, pro-affirmative action demonstrators bearing “Fight For Equality” placards marched on the sidewalk in front of the Supreme Court in a brisk wind. A parent-teachers group from Chicago and several civil-rights groups were among those sponsoring the demonstration.

Demonstrators chanted, “Equal education, not segregation” and “We won’t go to the back of the bus, integration is a must.” Some held signs that read “Stop racism now.” Among the crowd were representatives of the National Organization for Women, the NAACP, and students from Howard University.

Though outnumbered, there were some in the crowd from the other side.

“Regardless of how well-motivated, allowing the state to engineer racial mixing only creates racial stereotypes and increases racial tension,” said Terry Pell, president of the Center for Individual Rights, a public interest law firm. “The court needs to put an end to state-mandated tinkering with race.”

The school policies in contention are designed to keep schools from segregating along the same lines as neighborhoods. In Seattle, only high school students are affected. Louisville’s plan applies systemwide.

“The plan has prevented the resegregation that inevitably would result from the community’s segregated housing patterns and that most likely would produce many schools that might be perceived as ‘failing,'” the Seattle school district said in its brief to the high court.

The Bush administration has taken the side of the parents who are suing the school districts, much as it intervened on behalf of college and graduate students who challenged Michigan’s affirmative action policies in 2003.