The case turned on three design patents covering the appearance of early editions of the iPhone—including the device’s black rectangular shape and the layout of icons on the screen—which led a jury to order Samsung to pay $399 million in damages.

Before the Supreme Court, the issue at stake was not whether Samsung infringed on the patents, but instead how much the Korean company should pay based on a law that allows a patent owner to receive a competitor’s “total profit.” Should that profit be for the entire value of the smartphone, as an appeals court ruled, or only for profits attributable to the copied design?

In trying to make sense of the design patents’ value, the judges repeatedly invoked the body shape of Volkswagen’s iconic Beetle model, noting that consumers will pay extra for a cool-looking car. But they drew back at saying a company, in cases of complex products, should be able to use a patent for exterior appearance to collect for the whole thing.

Justice Stephen Breyer contrasted simple products like wallpaper to cars and smartphones, which are often covered by hundreds or thousands of patents and design decisions.

“For wallpaper, you get the whole thing. A Rolls Royce with the thing on the hood? No, no, no you don’t get profits on the whole car,” said Breyer.

Much of Tuesday’s argument was dedicated to the complicated question of how a jury should define a so-called “article of manufacturer” and how it should award damages. (Alas, none of the Justices revealed if they are iPhone or Android users—or if they use smartphones in the first place).

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Samsung’s lawyer, Kathleen Sullivan of Quinn Emanuel, argued in favor a two-part process in which a court would first establish the patented “article” in dispute, and then assess the damages based on expert witnesses or consumer survey evidence.

Apple’s lawyer, Seth Waxman of Wilmer Hale, devoted much of his time to a technical argument, saying the Supreme Court shouldn’t consider the matter at on the grounds that Samsung had failed to say at trial that the design patents did not cover the entire iPhone.

The Justices appeared to reject Waxman’s invitation, at times asking him to stop harping on the evidentiary record, and instead to focus on defining a process to define “articles of manufacture.”

Both Apple (AAPL) and Samsung (SSNLF) mostly agreed with a four-part approach to the definition process proposed by the Justice Department, which also appeared before the Supreme Court, and likewise rejected the lower court’s position that “total profit” should always cover the value of the entire product.

The hearing was unusual in that, by the time Apple and Samsung appeared for arguments, the parties had abandoned earlier positions on how to value design patents, and instead both agreed on Tuesday that the lower court’s “total profit” interpretation was wrong.

As a result, the Supreme Court’s decision will almost certainly overturn the lower court’s ruling, and send the case back to a trial court for a jury to determine the proper value of the design patents. As for what test the lower court’s should use, Justice Breyer repeatedly expressed support for a proposal by Internet Association, a trade group backed by Facebook (FB) and others, which said:

[W]here a design “has been applied” to only part of a multicomponent product and does not drive demand for the entire product, the “article of manufacture” is rightly considered to be only the component to which the design applies, and only profit attributable to that component may be awarded.

A ruling on the case is likely to come in December or January.

Both Apple and Samsung issued statements after the ruling, but ones that largely skirted the substance of the hearing.

“We firmly believe that strong design patent protection spurs creativity and innovation … Eleven times now, Samsung has been found guilty of intentionally and blatantly copying the iPhone. We think that’s wrong and that it poses chilling risks to the future of design innovation,” said Noreen Krall, Apple’s Chief Litigation Officer.

Samsung, for its part, published a statement that pointed to its own large portfolio of design patents, and called on the court to change the damages rule.

“Awarding all of the profits for a single patent devalues the contributions of the hundreds of thousands of other patents in a smartphone. We are hopeful that the Supreme Court will give a sensible and fair reading to the design patent statute.”

A multi-front battle

The dispute, which some are calling the “design case of the century,” involves just one aspect of a long-running feud between the companies that began in 2010 when Apple’s late CEO accused Samsung of “slavishly copying” the iPhone. The case has given rise to all sorts of legal issues, including the scope of patents and how other forms of intellectual property, such as trademarks and trade dress should apply.

Apple obtained the three patents in 2009 and 2010. Less common than “utility patents,” which protect new mechanical or industrial discoveries, design patents are intended to cover the distinct ornamental aspects of an invention.

The case saw a number of tech companies as well as an influential group of law professors support Samsung in calling for a more narrow interpretation of “total profit” in the case of design patents. Meanwhile, design-focused companies like Tiffany and Co (TIF) and Adidas weighed in to back Apple.

The hearing on Tuesday came less than a week after an appeals court reinstated a $120 million verdict in favor of Apple in a separate patent dispute—a ruling that Samsung has indicated it may likewise appeal to the Supreme Court.

This story was updated at 1:05pm Et to include the companies’ statements.