In order to remove 3 trapdoors per turn, both Beethro and the timeclone must step, and then the timeclone must push Beethro.
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×Since both are moving eastward (for the first half), the timeclone must step E or NE every turn. That implies he will push Beethro in the same direction.
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×So Beethro doesn't need to step eastward, as the push will keep his horizontal motion in step with the timeclone. His moves should be only north and south.
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50th Skywatcher