Brain Teasers
The Troll Bridge 1
A bridge was guarded by an evil troll. The troll was very intelligent, but he was also a coward. He was afraid of anyone smarter than him. So every time anyone tried to cross the bridge, the troll would set up a test. If the traveler passed the test, he would be allowed to cross. Otherwise, the troll would eat him.
Three travelers, Al, Ben and Carl, came across the bridge. The troll told them, "You may only cross my bridge if you know the password." He wrote five three-letter words on a rock. The five words were HOE, OAR, PAD, TOE, and VAT.
He then said, "I will tell each of you a different letter from the password. If you know what the password is, I will let you pass. But don't tell anyone else your letter." He then whispered a letter from the password to each traveler so that neither of the other two could hear him.
Then the troll asked Al, "Do you know what the password is?" "Yes," said Al, and the troll let him pass.
Then the troll asked Ben, "Do you know what the password is?" "Yes," said Ben, and the troll let him pass.
Then the troll asked Carl, "Do you know what the password is?" "Yes," said Carl, and the troll let him pass.
So, what is the password?
Three travelers, Al, Ben and Carl, came across the bridge. The troll told them, "You may only cross my bridge if you know the password." He wrote five three-letter words on a rock. The five words were HOE, OAR, PAD, TOE, and VAT.
He then said, "I will tell each of you a different letter from the password. If you know what the password is, I will let you pass. But don't tell anyone else your letter." He then whispered a letter from the password to each traveler so that neither of the other two could hear him.
Then the troll asked Al, "Do you know what the password is?" "Yes," said Al, and the troll let him pass.
Then the troll asked Ben, "Do you know what the password is?" "Yes," said Ben, and the troll let him pass.
Then the troll asked Carl, "Do you know what the password is?" "Yes," said Carl, and the troll let him pass.
So, what is the password?
Hint
1. Ben used Al's answer, and Carl used both Al's and Ben's answers.2. Al's letter could not have been E.
Answer
The password was HOE.The only letters unique to one of the five words were D, H, P, R, and V. One of these letters must have been Al's, since he could determine the password. The only word without any of these letters is TOE. So the password could not have been TOE.
Ben knew this and now had four words to think of. The only letters unique to one word were D, E, H, P, R, T, and V. The password must have contained at least two unique letters, one for Al and the other for Ben, since they were both able to determine the password. The only word with only one of these letters was OAR. So OAR could not have been the password.
Carl knew this and now had three words to think of. The only letters unique to one word were D, E, H, O, P, T, and V. The password must have included three of these letters, since all three were able to determine the password. The only one of the three words containing three such letters was HOE, so this must have been the password.
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Comments
I love this puzzle! The logic is pretty straightforward, but keeping everything in your head is quite a challenge. And a hungry troll always adds to any brain teaser! Wonderful.
A scip model for solving this puzzle can be found at
https://gist.github.com/saska-gist/
7fb42e0fdcf64aeb8ce6d7e2c6f94e17
https://gist.github.com/saska-gist/
7fb42e0fdcf64aeb8ce6d7e2c6f94e17
I don't like this at all. Because the puzzle as written does not allow a unique answer. Suppose the password was PAD and the troll whispers P to A, then whispers D to B, then whispers A to C.
Then A is asked "Do you know the password?" "Yes", says A, because he had a unique letter.
Then B is asked "Do you know the password?" "Yes", says B, because he had a unique letter.
Then C is asked "Do you know the password?" "Yes", says C, because A and B both were able to answer "Yes", and there is only one word which contains an A that allows for that to happen.
The fact that B can hear that A says "Yes", and that C can hear that A and B both said "Yes" provides a different piece of information that affects the result and means that the problem no longer has a unique solution.
Then A is asked "Do you know the password?" "Yes", says A, because he had a unique letter.
Then B is asked "Do you know the password?" "Yes", says B, because he had a unique letter.
Then C is asked "Do you know the password?" "Yes", says C, because A and B both were able to answer "Yes", and there is only one word which contains an A that allows for that to happen.
The fact that B can hear that A says "Yes", and that C can hear that A and B both said "Yes" provides a different piece of information that affects the result and means that the problem no longer has a unique solution.
Actually it does. In the setting you described. C would not be sure if the answer is VAT or PAD. Hence C cannot say "Yes" in that case.
The fact that C says YES, reverses us to the conclusion that C knew a letter which was NOT the letter A.
The fact that C says YES, reverses us to the conclusion that C knew a letter which was NOT the letter A.
Wow! As of today this is the highest rated "Fun" teaser on the site. Yay for logic puzzle fans! Congrats and thanks WimpyKid for bringing this to us.
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