I have a pet ant that I keep on a leash. I keep her on the outside surface of a cube. Her leash is twice the length of an edge of the cube. I'm trying to decide where to attach the leash.
A: At a vertex.
B: At the center of an edge.
C: At the center of a face.
Which choice gives my pet ant the most area to roam?
Ok, I think I have the solution but my gut is telling me I'm getting something wrong, I just can't seem to figure out where the flaw is in my logic, so hopefully one of you will be so kind as to set me straight :-).
Consider an infinite grid of squares on a plane. This can be used to represent the faces of the cube. If you draw a line on this grid that will represent the leash, anytime it crosses a line of a square that represents the leash wrapping over an edge. Using this reasoning then the total area reachable by the ant is simply a circle centered on the ant with a radius of the length of the leash. Thus no matter where you place the ant it can reach the same area. Thus all 3 are equal.
To help further explain my reasoning, I'm borrowing the "trick" used in billiards to determine the path of a ball as it bounces off the walls, you simply copy the table as a grid and each the path of the ball crosses is a line it represents a reflection, only difference here is that the line crossing represents a face crossing instead of a reflection.
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Posted by Daniel
on 2018-08-31 08:46:45 |
Some Trouble (possible solution)
