If you travel 1/m way to the mth vertex (the first vertex being 1, and thus, you start there), picking the mth vertex in any order such as to not repeate (waves hands all over the place and even starts to fly), you wind up at the center of mass of that polygon.

Since there can only be one center of mass, you always wind up in the same spot.

]]>Edit: Center of mass of a polygon with equal density in all places, that is.

]]>A treasure map has n villages marked on it, and it contains the following instructions: Start at village A, go 1/2 of the way to village B, 1/3 of the way to village C, 1/4 of the way to village D, and so forth. The treasure is buried at the last stop. Problem: You lose the instructions, and don't know in what order to select the villages. Prove that the order you select the villages in doesn't matter.

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