We know there are six messages, only one person can leave more than one message, and the most messages any one person can leave is three. That means the absolute minimum number of callers is four (one calls three times, the remaining three each call once, giving us a 3-1-1-1 distribution).
If G leaves a message, F and P do. If F leaves a message, P and T do. If P leaves a message, H and L do. Let’s represent these relationships as:
G ? F ? T
G ? P ? H & L
And note the contrapositives
NO T ? NO F ? NO G
NO H or NO L ? NO P ? NO G
If T were out, then we couldn’t have F or G either. That would be three people out, leaving just three people to call. But as we said, the minimum number of callers is four. So, T can’t be out.
If H or L were out, then we couldn’t have P or G either. Again, that would be three people out, leaving just three people to call. Again, that wouldn’t let us get to our minimum number of callers. So H and L can’t be out, either.
