# The Secret Word

Here is a new logic puzzle from Presh Talwalkar. A teacher has a puzzle for three students: Alicia, Brittany and Cheryl. The teacher puts the following words on the board: CAT, DOG, HAS, MAX, DIM, TAG.

The teacher then hands each student a slip of paper with a single letter written on it, explaining that the three letters together spell one of the words on the board. The teacher asks Alicia if she knows what the word is, and she answers Yes. Then the teacher asks Brittany if she knows what the word is, and she thinks for a moment and also answers Yes. Finally she asks Cheryl if she knows what the word is, and she also thinks for a moment and says Yes. What is the word?

If you want a hint, click on See Explanation below and read as far as you want to. If you think you have figured out the answer, click See Answer.

See Explanation
One approach to figuring this out is to start with Alicia. She knew the word before hearing anything from the others, so her letter must be unique. The only way Alicia could know the word right away is if the letter on her piece of paper is C, O, H, S, X or I. All the other letters appear more than once, so if Alicia had one of those letters, she would have answered No. This eliminates the word TAG because it has no unique letters. So from the fact that Alicia knows the word, Brittany and Cheryl now know that the word is not TAG, even without looking at their letters.

Now think about Brittany. She also answered Yes, after looking at her own letter and thinking about Alicia’s answer. So Brittany’s letter, combined with one of Alicia’s letters, must narrow it down to one word. This eliminates two possibilities:

• The word cannot be MAX. If Alicia had X, and Brittany had M, Brittany would not be sure what the word is. From Brittany’s point of view, all she would know is that she has M, and Alicia 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 Brittany had M, the same logic that eliminated MAX would apply: the word could be MAX or DIM and Brittany would not be able to answer Yes. If Brittany had D, from her point of view Alicia could have O or I, the word could be DOG or DIM, and she could not answer Yes.

All this can be figured out by Cheryl 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 Cheryl’s letter, and what is the word?

• We can eliminate HAS. If Alicia had H and Brittany had S, or vice versa, then Cheryl would have to have A. But if Cheryl 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 Alicia had C and Brittany had T, then Cheryl would have to have A, but she would not know whether the word was CAT or HAS.

Therefore…