Final answer:
The Supreme Court case that found that states may incarcerate offenders only if they have been represented by counsel is Argersinger v. Hamlin, in which the Court held that the Sixth Amendment's right to counsel extends to felony defendants in state courts.
Step-by-step explanation:
The Supreme Court case that found that states may incarcerate offenders only if they have been represented by counsel is Argersinger v. Hamlin. In this case, the Court held that the Sixth Amendment's right to counsel extends to felony defendants in state courts. The Court determined that the right to counsel was essential for a fair trial, and therefore, states must provide attorneys for defendants who cannot afford their own.