Checking the Doors

36 square rooms are arranged in a 6x6 grid.  The outside wall of the grid has no doors.  All other walls have exactly one door, for a total of 60 doors.  A guard follows a path that takes him through as many doors as possible, never going through the same door twice (he may enter a room twice, as long as he uses different doors.)  What is the maximum number of doors he can use if:

  1. His path must be circular (must start and end in the same room)?
  2. His path may start and end in different rooms?

Extensions: Repeat for 3x3 to 8x8 grids.

Source: Original.

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