"He stopped playing."
Translation:Αυτός σταμάτησε να παίζει.
I think, the verb before the να indicates the time (present, past) the verb after the να indicates the aspect (repetition, only once) σταμάτησε να παίζει = he stopped playing (forever) σταμάτησε να παίξει = he stopped playing (the current game) σταμάτα να παίξει = he stops playing (the current game)
any original greek to verify this understanding?
(1) Παίζει = he plays / he is playing: this it shows an action that is happening, that has duration. In the future tense, θα παίζει = he will be playing. When in the subjunctive, να παίζει, it shows that he stopped the action that was in progress, so he stopped playing.
(2) In παίξει, the -ξ- notes an instant: θα παίξω = I will play, but not I will be playing (that is θα παίζω, see above). Σταμάτησε για να παίξει means he stopped in order to play, ie he stopped whatever he was doing, which we do not know, in order to play.
You can't say σταμάτησε να παίξει. :) It's either (1) stopping the action in progress or (2) starting a new one which is denoted with για= in order to and then the perfective aspect.
You can review the conjugation table for παίζω here.