# Speed Limits on a Cube

Geometry Level 5

A $2014\times 2014\times 2014$ 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 $1$ unit per minute. If the bug is on the left or right face, then it drives at $2$ units per minute. Finally, if the bug is on the front or back face, it drives at $3$ 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 $M$. Find the value of $\lfloor M\rfloor \pmod{1000}$

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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