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We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in thismanner a characterization of amenability in terms of cellular automata: "Agroup $G$ is amenable if and only if every cellular automaton with carrier $G$that has gardens of Eden also has mutually erasable patterns."  This answers a question by Schupp, and solves a conjecture byCeccherini-Silberstein, Mach\`i and Scarabotti.  An appendix by Dawid Kielak proves that group rings without zero divisors areOre domains precisely when the group is amenable, answering a conjectureattributed to Guba.

Amenability of groups is characterized by Myhill's Theorem