A double shine (or multi shine, triple shine, etc.) is the repeated use of Reflector as an attack by either Fox or Falco. It is performed by jump-cancelling a Reflector and using a second Reflector just after leaving the ground, timed so that Fox or Falco will quickly land and allow the pattern to be repeated.
The Ice Climbers is a character in which the player controls Popo (blue IC depending on costume color), while Nana follows and mimics Popo's inputs with a slight delay. The main strategy with Ice Climbers is abusing their grab combos, as they can kill off of most grabs. Strategies change constantly with IC's based on if your Nana is present or not.
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