Sobot or fauxbot?

Question

Through the bustling hallway, you followed your speed-walking, speed-talking brother, who was hoping to submit an article about the upcoming demonstration to his beloved Logic and Philately Semi-Quarterly.

"There are two kinds of robots here," he explained: "those who only make true statements -- the 'sobots' -- and those who only make false statements -- the 'fauxbots'. They are also all perfect logicians; so in this demonstration, five (who know what each other are) will be witnesses and one (who doesn't know what the others are) will sit as judge, and try to determine from the statements the witnesses make which of them are sobots and which are fauxbots."

"Couldn't the judge just ask them, I don't know ... 'Is two plus two four?'"

Your brother made a pooh-poohing gesture. "That would be trivial -- and moreoever, in bad taste. It has become the custom among robots to never ask another robot, directly or indirectly, if they are honest or a liar, but to deduce it from the things they say. It never takes a robot long to figure it out, anyway, and then they know exactly where they stand: since a sobot CAN'T tell a lie and a faubot CAN'T tell the truth, they know the truth once they know who they're talking to -- which is more than we humans can say!"

Following him closely, you pushed your way into the overcrowded classroom in which the demonstration was taking place. From your spot at the back of the room, you could neither see nor hear anything happening on the dais at the front. Your brother, who was a head taller than average, had somewhat better luck, and reported back to you what he was able to make out.

"It's just started," he said. "The fourth one just pointed to the other four and said that one of them is something something -- I couldn't hear. It sounded like 'a robot'; it must have been 'a sobot' or 'a fauxbot'."

A fellow observer shushed your brother; he made a face and continued to whisper. "Now the first one just pointed at the second, third, and fifth ones, and said that one of them is something or other -- a sobot or a fauxbot."

And a few moments later: "The third one just said that the first and the fourth are both something -- both sobots or both fauxbots."

And then: "The first one just said something about the third one -- that he's a sobot or a fauxbot, presumably. Agh, this is so frustrating!"

And then: "The first one just pointed at the third and fourth and said they are both sobots or fauxbots -- I couldn't hear which."

And then: "The fourth one just said that both the first one and the third one are sobots or fauxbots."

"That's not enough!" someone cried. ("That's the judge," whispered your brother.) "That I now have everything I need to render judgement and that I am a sobot -- are both untrue!"

The rumbling hubbub settled briefly to a murmurous drone; but as the judge announced the verdict, the audience's reactions drowned it out for those standing farther away; all you and your brother could hear was this:

"The first is not ... The second is not ... The third is not ... The fourth is not ... And finally, the fifth is not ... Thus concludes the demonstration. Please clear the room!"

In rapturous amazement, the crowd dispersed tumultuously, driving you and your brother out into the hallway.

"What a pain!" he complained. "I could hardly hear anything. I have no idea who said what, who was what, or whether the judge's judgement was impressive -- or even correct!"

"It wouldn't have been correct," you mused. "The judge was obviously a fauxbot."

"Yes, yes; but you know what I mean. Oh, it's a crying shame about that article ..."

You condoled absentmindedly, and soon your brother took his leave.

An hour later you called him. "Good news about that article," you said. "If you were right about who talked about whom, and that the judge is indeed an infallible logician, then I know which of the witnesses were sobots and which fauxbots -- and moreover, what each of them testified."

"What!? How!?"

What did you know, and how?

Answer 1

You know that

all of the robots are fauxbots.

This is because

There are 32 possible permutations of the five robots. Right off the bat, 17 of them are impossible - for example, it's not possible for 1 and 3 to be sobots, and 4 to be a fauxbot, because then your brother could not have said "The first one just pointed at the third and fourth and said they are both sobots or fauxbots -- I couldn't hear which."

Of the fifteen remaining permutations, you conclude what the inaudible parts of the testimony would have been, e.g. consider the possibility that 2 is a fauxbot and the rest are sobots. Then we know what each of the statements were (since 1, 3, and 4 are the only ones who spoke, and they would have all spoken the truth). But this would have been indistinguishable from the case where 5 is a fauxbot and the rest are sobots - all of their statements would have been the same, and the judge would not have been able to logically deduce the identity of robots 2 and 5.

As it turns out, in only one of the fifteen potential permutations would the judge have been able to determine the truth following the sixth statement. This is the case when the second statement ends in "fauxbot" and the rest end in "sobot." More explicitly:

Robot 4 points to the other four and says, "One of them is a sobot."
Robot 1 points to 2, 3, and 5 and says, "One of them is a fauxbot."
Robot 3 points to 1 and 4 and says, "They are both sobots."
Robot 1 points to 3 and says, "He is a sobot."
Robot 1 points to 3 and 4 and says, "They are both sobots."

After hearing these five statements, the judge has determined that it's possible that robot 4 is a sobot, in which case one of robots 2 or 5 is also a sobot and the rest are fauxbots. Or it's also possible that all five of the robots are fauxbots. He does not have enough information at this point to determine which of those is the case, however.

Finally, robot 4 points to 1 and 3 and says, "They are both sobots."

At this point the judge knows that robot 4 is lying, and therefore all five robots are fauxbots.