In which case did the offender successfully argue against the death penalty due to "cruel and unusual punishment"?

In the context of challenging the death penalty based on the Eighth Amendment's prohibition of cruel and unusual punishment, the significant case is Furman v. Georgia. In this landmark decision, the Supreme Court ruled that the death penalty, as it was administered at the time, was arbitrary and capricious, leading to a constitutionally unacceptable risk of discriminatory application. The ruling effectively put a halt to capital punishment across the United States, as the Court found that it violated the Eighth Amendment due to its inconsistency and potential for severe injustice.

The other cases listed address different aspects of the death penalty and punishment. Gregg v. Georgia later upheld the constitutionality of the death penalty with certain reforms to address previous concerns, while Baze v. Rees focused on the method of execution rather than the death penalty itself. Miller v. Alabama pertains to mandatory life without parole for juveniles, emphasizing the developmental differences in youth rather than challenges to the death penalty.