A robot sits at the origin of a two-dimensional plane. Each second the robot chooses a direction, North or East, and at the th second moves units in that direction. The total number of moves made by the robot is a multiple of 3. Show that, for each possible total number of moves, there are at least four different routes the robot can take such that the distance from the origin to the robot''s final position is an integer.
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Maclaurin Mathematical Olympiad (2022)
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