Munchkin

Does the "reasonable time rule" apply to defeating a monster without killing it, or just to killing it?

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Any sort of defeat. If you defeat it without killing it, other players still have a reasonable time to play an applicable card to frustrate you. However, if you removed the only monster from the fight (with Magic Lamp, for instance), the fight is over. Your opponents cannot play monster enhancers or other similar cards that affect a specific monster, because there isn't one there to fight, and they can't use any special rules for bringing in monsters (such as the Shark rules from Munchkin Booty) that depend on having a monster in the fight.
Related Rule(s)