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Adults do not play the Nash equilibrium in the well known centipede game. While Palacios-Huerta and Volij (2009) argued that behavior results from the failure of backward induction logic, Levitt et al. (2011) found that players who know how to backward induct still do not play Nash. Here, we ask children and adolescents (ages 8 to 16) to play the centipede game in the laboratory and we leverage knowledge about developing abilities to assess the contribution of backward induction logic. In line with the literature, we find that the ability to perform backward induction increases with age. However, it predicts behavior only in elementary school children: those with advanced logical abilities over-apply their skills. Starting in middle school, students who reason logically know that the unraveling argument should not be applied blindly. They utilize Theory-of-Mind (ToM) abilities to form beliefs about others' play and (optimally) refrain from stopping immediately. Their behavior is in line with the deviations observed in adults. Interestingly, developing ToM leads to a gradual decrease in stopping stages with age, which is accompanied by a decrease in payoffs with age. The results indicate that ToM is the key contributor of behavior that helps departing from backward induction when beneficial.