The justices' most pressing task is to resolve the fate of the federal criminal sentencing system, which the court itself threw into limbo in June by declaring unconstitutional a similar, although not identical, system used by the state of Washington.

In both the state and federal systems, sentencing guidelines provide a starting point for calculating a criminal sentence, and judges then make findings about a variety of factors to determine how much time a defendant will actually serve. The Supreme Court held in Blakely v. Washington that the state system violated the constitutional right to trial by jury by permitting judges to make these essential findings.

Federal judges around the country quickly started ruling that they could no longer treat the federal sentencing guidelines as binding. Whether that judgment is correct and, if so, what should happen next will be the subject of an unusual afternoon argument on Monday in two cases that the justices granted in August at the Justice Department's request and agreed to expedite for an argument that would not ordinarily have been scheduled until January.