### Brain Teasers

# A Prisoner

A prisoner sat alone in his cell staring through the bars at the six doors to the guardroom beyond them. Suddenly, there was the sound of footsteps. One guard came in through door A and a second entered through door F. They unlocked the cell door. "We shall set you free," said one of the guards, "but first 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; B by C or E; C by D or F; D by A or F; E by B or D; and F by C or D." With that the guards left through door B. In what order must the prisoner pass through the doors?

### Hint

The prisoner must alternate exits and entrances. List the six possible orders then study the conditions to eliminate possibilities.### Answer

The order is CFDABE. The guards entered through doors A and F and exited at door B. The prisoner 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 until the prisoner has completed his task and finally wins his freedom through B or D!

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