Climax is best defined as the "turning point in a storyline," the moment that the rising action turns into the falling action (Literary Devices, "Climax"). At this moment, the conflict in the narrative reaches its greatest intensity, bringing the resolution into sight. Although some literary critics use the terms crisis and climax interchangeably, "the climax usually follows" but can sometimes overlap with the crisis (Dr. Wheeler, "Literary Terms and Definitions: C"). At the point of crisis, the reader is still unsure what the resolution will be, if the protagonist will succeed or fail in battling the conflict. Hence, to find the climax, we first want to determine the resolution.
In Richard Edward Connell's short story "The Most Dangerous Game," we know the story resolves with Rainsford deciding "he had never slept in a better bed" than General Zaroff's, which means that Rainsford won the battle between he and Zaroff in Zaroff's bedroom. Hence, we also know that the climax comes just prior to this moment.
Before Zaroff has dinner that night and locks himself in his bedroom, the last thing we witness Rainsford doing is standing on the beach, staring into the sea, and hearing the dogs drawing closer. The narration in the paragraph ends with, "Then he leaped far out into the sea ...." At this point, the reader is not completely sure what happened to Rainsford. It seems as though he may have committed suicide in order to escape being caught by the dogs and Zaroff. Since the resolution of the story is uncertain at this point, we know that this is the moment of crisis in the story and that the climax is soon to follow.
Hence, the climax is the moment that Zaroff locks himself in his bedroom, goes to his bedroom window to admire the moonlight on the courtyard, and turns around to see Rainsford, "who had been hiding in the curtains of the bed," standing there. At that moment, we know that Rainsford has outwitted Zaroff and won the game; all that's needed next is the final battle. Since at this moment the resolution is already in sight, we know this moment is the climax.
In addition, The conversation in which Zaroff declares Rainsford has won and in which Zaroff realizes Rainsford intends to fight Zaroff to the death counts as the falling action that leads to the resolution.