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Increasing fetch as sea ice retreats with global warming is increasing the amplitude of ocean waves, motivating the need for a better understanding of the impact of episodic flexing on the strength of ice. Unexpectedly, recent studies showed that the flexural strength of ice increases by as much as a factor of two or more upon cyclic loading (unlike earlier results where ice had a thermo-mechanical history that could account for the difference), possibly owing to the development of an internal back stress originating from dislocation pileups. New systematic experiments reveal that the cyclically-induced increase in flexural strength of columnar-grained S2 freshwater and saline ice is fully relaxed upon annealing at high homologous temperatures (T_h=0.91 and 0.96). Moreover, the ice can be repeatedly strengthened to the same level by cyclic loading if allowed to anneal after each episode of strengthening. The relaxation of the original strength is attributed to the relaxation of the cyclically-induced internal back stress.