The number reduction task (NRT) allows us to study the transition from implicit knowledge of hidden task regularities to explicit insight into these regularities. To identify sleep-associated neurophysiological indicators of this restructuring of knowledge representations, we measured frequency-specific power of EEG while participants slept during the night between two sessions of the NRT. Alpha (8-12 Hz) EEG power during slow wave sleep (SWS) emerged as a specific marker of the transformation of presleep implicit knowledge to postsleep explicit knowledge (ExK). Beta power during SWS was increased whenever ExK was attained after sleep, irrespective of presleep knowledge. No such EEG predictors of insight were found during Sleep Stage 2 and rapid eye movement sleep. These results support the view that it is neuronal memory reprocessing during sleep, in particular during SWS, that lays the foundations for restructuring those task-related representations in the brain that are necessary for promoting the gain of ExK.