A teacher writes six words on a board:
CAT DOG HAS MAX DIM TAG
She gives three students Abner, Bentley, and Charlotte each a piece of paper with precisely one letter from one of the words, so that:
- Each student gets a different letter, and:
- All three students get a letter from the same word.
Then she asks, "Abner, do you know the word?"
Abner immediately replies "yes".
She then asks, "Bentley, do you know the word?"
Bentley thinks for a moment and replies "yes".
Then she asks Charlotte the same question.
Charlotte thinks for a short while and then replies "yes".
What is the word?
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Submitted by K Sengupta
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Rating: 5.0000 (1 votes)
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Solution:
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One approach to figuring this out is to start with Abner. He knew the word before hearing anything from the others, so her letter must be unique. The only way Abner could know the word right away is if the letter on his piece of paper is C, O, H, S, X or I. All the other letters appear more than once, so if Abner had one of those letters, he would have answered No. This eliminates the word TAG because it has no unique letters. So from the fact that Abner knows the word, Bentley and Charlotte now know that the word is not TAG, even without looking at their letters.
Now think about Bentley. He also answered Yes, after looking at his own letter and thinking about Abner’s answer. So Bentley’s letter, combined with one of Abner's letters, must narrow it down to one word. This eliminates two possibilities.
The word cannot be MAX. If Abner had X, and Bentley had M, Bentley would not be sure what the word is. From Bentley’s point of view, all he would know is that he has M, and Abner has a unique letter. But that could be X or I, and the word could be MAX or DIM.
The word cannot be DIM either. If Bentley had M, the same logic that eliminated MAX would apply: the word could be MAX or DIM and Bentley would not be able to answer Yes. If Bentley had D, from his point of view Abner could have O or I, the word could be DOG or DIM, and he could not answer Yes.
All this can be figured out by Charlotte without even looking at her own letter. She now knows that the word must be CAT, DOG or HAS. She knows that Alicia’s letter is C, O, H or S. Now what is Charlotte's letter, and what is the word?
We can eliminate HAS. If Abner had H and Bentley had S, or vice versa, then Charlotte would have to have A. But if Charlotte had A, she would not be able to answer Yes, because the word could still be CAT.
We can eliminate CAT by the same logic. If Abner had C and Bentley had T, then Charlotte would have to have A, but she would not know whether the word was CAT or HAS.
Therefore…
The secret word is DOG. |
