Puzzle – Secret Word

A teacher writes six words on a board: “cat dog has max dim tag.” She gives three students, Albert, Bernard, and Cheryl each a piece of paper with one letter from one of these above written words on the board. 

Then she asks, “Albert, do you know the word?” Albert immediately replies yes. 
She asks, “Bernard, do you know the word?” He thinks for a moment and replies yes. 
Then she asks Cheryl the same question. She thinks and then replies yes. What is the word?

Puzzle – Secret Word

Solution:

We’re going to analyze the letters in each of the words in relation to the responses that Albert, Bernard, and Cheryl give. 

The teacher first asked if Albert knows the word and he responds that he does know the word. This means that Albert’s letter can uniquely identify a word. In other words, the frequency of the letter that he got is exactly one. 

For example, consider if Albert had gotten the letter ‘c’. The letter ‘c’ only appears in the word cat, so if Albert had gotten the letter ‘c’ he would be able to figure out that the word is a cat. In contrast, suppose Albert had gotten the letter ‘a’. the letter ‘a’ appears in multiple words, so there would have been no way for Albert to figure out which word was the word that they got. Since Albert replied he was able to identify the word that means he did not get the letter A so we need to find out which letters appear exactly once in this list of words.
We can go through all of the words and we can figure out which of these letters uniquely appear one time in this list. These are the letters: c, o, h or s, x, or i.

Albert’s Answer Justification

Notice in the word tag that the letters t, a, and g all appear multiple times in this list. So since Albert replied that he could identify the word, that means he didn’t have any of these letters. The word cannot be TAG.

The teacher then asked Bernard if he knows the word. Bernard, after a moment of thinking, also replies yes so Bernard has heard Albert’s answer and wants to continue the logic, so from the remaining list of words, Bernard would be able to uniquely identify which word they all had. so we go through this list. It could be the letters t, g, h, or s. These are the possible choices the other letters appear more than once in this list of remaining letters that Bernard could have gotten. What this means is that the word could not be ‘max’ or ‘dim’ so we can eliminate those words. 

Bernard’s Answer Justification

Now Cheryl, who has heard all of these, is then asked if she knows the word. She looks at her letter and thinks about what Albert & Bernard said and also says, “Yes, I can figure out what the word is.” So, continuing the logic, one of the remaining letters has to uniquely identify the word. Well, there are only two letters left there’s a letter a and there’s a letter d. The unique letter left is letter d because the letter a appears in both the words ‘cat’ and ‘has’ so Cheryl must have gotten the letter D and it could not have been the word cat or has.

Cheryl’s Answer Justification

So, we can conclude that Albert would have gotten the letter O, Bernard would have gotten the letter G, and Cheryl would have gotten the letter D.

So, the secret word has to be ‘dog’.