HAS does in fact have two unique letters, but let’s consider the scenario of person 1, 2, and 3 having the letters C, T, and A, respectively. Person 1 has C, so they know the word is CAT. Person 2 has T, so they know the word can be either CAT or TAG. But If the word were TAG, person 1 would not have claimed to know the word, since person 1 needs to hold a unique letter to know. Therefore, person 2 having a T knows the word is CAT. Then, person 3 sees that the first two people knew the word. Person 3 has an A, but note how this is the exact same situation as the HAS scenario. So person 3 cannot know if the first two people held an H and S, or if they held a C and T. That’s why HAS cannot be the word.

Following this logic, one can deduce that the only possible working scenario is DOG, with person 1 having the O, person 2 having the G, and person 3 having the D.