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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:

- His path must be circular (must start and end in the same room)?

- His path may start and end in different rooms?

Extensions: Repeat for 3x3 to 8x8 grids.

Source: Original.

Solution

Mail to Ken