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Suppose I have two identical baseballs. I mark one with a black dot. I use two ball launchers to put them on a collision course with each other. The marked ball is in the left launcher, the unmarked ball in the right launcher. I tell you everything about the launch speeds, angles, positions, etc. They collide behind a barrier that I can't see around. But then they reappear, one to the left and to the right after the collision behind the barrier. The game is to predict which baseball has the black dot on it and then go look at the two balls and see if you're right.
I think we agree that you can always win this game.
Now repeat this experiment with two identical atoms. Under what circumstances (if any) can I still win this game?