Doorman Riddle
Once upon a time, two doormen were guarding the royal palace portal to the King's chamber. Whoever would have an audience with his royalty must pass and exit through this door only. One day, there appeared visitors all with different ranks, similar to those in chess hierarchy (from highest to lowest respectively): the Queen, the Bishop, the Knight, the tower guard, and the foot soldier.
The first to enter this door simply walked through while these doormen were not allowed to look inside the palace. Then the following visitor would wait outside until the previous visitor finished the business with the King and called out for an exit. When the consecutive visitors met upon door opening, whoever with the lower rank must bow while the higher one needed not. Nonetheless, once again, the doormen could not see the inside and would only see someone bowing from the outside only.
Doorman #1 : Of this order, we saw alternating acts of passing. The first one just walked through. Then the second bowed while the third did not. Then the fourth one bowed, and the last did not.
Doorman #2 : Indeed. And even if the middle visitor was absent, of this order, we would see those who bowed would still need to bow.
Doorman #3 : O, so true. And even if those who bowed were absent from this order, we would see the remaining three who had not bowed still needed not to bow at all.
What was the order (from first to last) of these visitors? (Let stand for the lowest to the highest ranks respectively, as mentioned above. For example, 5 stands for the Queen, and 1 stands for the foot soldier.)
The answer is 32415.
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1 solution
From the question, let 5 stands for the Queen, 4 for the Bishop, 3 for the Knight, 2 for the tower guard, and 1 for the foot soldier.
Also, let the order of visitors be a , b , c , d , e from the first to last, respectively.
From the doormen's perspective, those who bowed mean those with rank numbers less than the previous ones.
That is, a > b < c > d < e .
In order words, a , c , e are of higher ranks and so can't be 1 . Thus, either b or d is 1 , and from the second constraint, where c is absent and the bowing sequence remains, we can deduce that a > b > d < e . That leads to b > d , so d = 1 .
As a result, b either 2 or 3 because 4 . 5 can't be lower than two other ranks.
However, if b = 3 , then a , c must be of 4 , 5 ranks to make the inequality true, making e = 2 , which is contradicted because from the last constraint, with b , d absent, it must follow a < c < e (no-one bowed from the doormen's perspective) but e = 2 < a , c .
Thus, e can't be 2 , and b can't be 3 .
Therefore, b = 2 , and the only way to write a < c < e is 3 < 4 < 5 . Finally, the order is 3 2 4 1 5 .
Literally, the visiting order from the first to last was as followed: the Knight, the tower guard, the Bishop, the foot soldier, and the Queen.