The Six Doors

Puzzle:

You are in a room that has six doors. You must pass through all six doors, each door once only, in the correct order. Three are entrances only and three are exits only. Door A must be followed by door B or E; D by A or F; B by C or E; C by D or F; E by B or D; and F by C or D. Also A, F are entrance and B is exit. In what order you must pass through the doors?

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Solution:

The order is CFDABE.

Doors A, F are entrance and B is exit. We must alternate exits and entrances. Let the unknown entrance be X. With a "_" denoting an exit door the possible orders are:
(a) _A_F_X ; (b) _F_A_X ; (c) _X_A_F ; (d) _X_F_A ; (e) _A_X_F ; (f) _F_X_A

Neither (a) nor (c) are possible since A is followed by B or E and neither leads to F. Since A can only be preceded by D, (e) and (f) are possible as DA_X_F and _F_XDA, respectively. With D used, door F can only be preceded by C to give DA_XCF and CF_XDA, respectively. Door B must occupy the remaining space in each possibility (third door is exit) so X=E but C cannot follow E in the former and for the latter, B cannot follow F. This eliminates (e) and (f). Since only D precedes A, (b) is possible as _FDA_X and (d) as _X_FDA. For (d), the exit B must occupy a "_" but cannot be followed by F, and if it is the first door, neither the C or E that must follow as X can be followed by the remaining E or C, respectively. This only leaves (b) and only C (of the possible B,C,E) can precede F as CFDA_X and B must occupy the final "_" as an exit leaving E=X. Notice that the condition that E is followed by B or D cannot be satisfied.