Speed Limits on a Cube

Geometry Level 5

A 2014×2014×20142014\times 2014\times 2014 cube has a little bug in a little car driving on its surface. If the bug is on either the bottom or top face, then it drives at 11 unit per minute. If the bug is on the left or right face, then it drives at 22 units per minute. Finally, if the bug is on the front or back face, it drives at 33 units per minute. Let the least amount of time in minutes it needs to drive from one vertex of the cubical world to the opposite vertex be MM. Find the value of M(mod1000)\lfloor M\rfloor \pmod{1000}

Details and Assumptions

The bug cannot drive along any of the edges of the world.

Wolfram Alpha might be necessary at the last step. (Sorry I couldn't make the numbers nicer!)


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